Search: keyword:new
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A372327
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Decimal expansion of Pi^(1/2)*Gamma(1/18)/(9*Gamma(5/9)).
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2, 1, 4, 9, 9, 9, 5, 4, 5, 8, 4, 9, 2, 0, 4, 7, 2, 3, 3, 9, 1, 2, 2, 2, 9, 4, 5, 6, 6, 3, 6, 5, 0, 8, 7, 5, 6, 3, 8, 7, 4, 8, 3, 1, 5, 1, 5, 7, 3, 7, 7, 8, 7, 9, 5, 6, 1, 7, 4, 7, 2, 8, 0, 3, 9, 8, 5, 7, 2, 7, 3, 5, 9, 2, 5, 4, 1, 7, 4, 9, 6, 1, 0, 4, 4, 4, 3, 5, 7, 5, 0, 0, 8, 3, 9, 7, 7, 8, 6, 5, 2, 6, 9, 6, 6, 9, 6, 8, 9, 2, 8
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OFFSET
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1,1
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COMMENTS
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Constant from generalized Pi integrals: the case of n=18.
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LINKS
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FORMULA
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Equals 2*Integral_{x=0..1} dx/sqrt(1-x^18).
Equals Gamma(1/18)^2 / (9 * 2^(8/9) * Gamma(1/9)). - Vaclav Kotesovec, Apr 29 2024
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EXAMPLE
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2.14999545849204723391222945664...
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MATHEMATICA
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RealDigits[Sqrt[Pi]/9*Gamma[1/18]/Gamma[5/9], 10, 5001][[1]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A370354
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Composite numbers k that share a factor with sopfr(k), the sum of the primes dividing k, with repetition.
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4, 8, 9, 16, 18, 24, 25, 27, 30, 32, 36, 42, 49, 50, 60, 64, 66, 70, 72, 78, 81, 84, 98, 100, 102, 105, 110, 114, 120, 121, 125, 126, 128, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 169, 170, 174, 180, 182, 186, 190, 192, 195, 196, 200, 204, 216, 220, 222, 228, 230, 231, 234, 238, 240
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OFFSET
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1,1
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COMMENTS
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All prime powers, see A246547, are terms.
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LINKS
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EXAMPLE
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18 is a term as 18 = 2 * 3 * 3, and soprf(18) = 2 + 3 + 3 = 8, which shares a factor with 18.
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A372524
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The smallest composite number k that shares exactly n distinct prime factors with sopfr(k), the sum of the primes dividing k, with repetition.
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OFFSET
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0,1
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LINKS
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EXAMPLE
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a(0) = 6 as 6 = 2 * 3 while sopfr(6) = 5, which shares 0 distinct prime factors with 6.
a(1) = 4 as 4 = 2 * 2 while sopfr(4) = 4 = 2 * 2, which shares 1 distinct prime factor, 2, with 4.
a(2) = 30 as 30 = 2 * 3 * 5 while sopfr(30) = 10 = 2 * 5, which shares 2 distinct prime factors, 2 and 5, with 30.
a(3) = 1530 as 1530 = 2 * 3^2 * 5 * 17 while sopfr(1530) = 30 = 2 * 3 * 5, which shares 3 distinct primes factors, 2, 3 and 5, with 1530.
a(4) = 40530 as 40530 = 2 * 3 * 5 * 7 * 193 while sopfr(40530) = 210 = 2 * 3 * 5 * 7, which shares 4 distinct prime factors, 2, 3, 5 and 7, with 40530.
a(5) = 37838430 as 37838430 = 2 * 3^2 * 5 * 7 * 17 * 3533 while sopfr(37838430) = 3570 = 2 * 3 * 5 * 7 * 17, which shares 5 distinct prime factors, 2, 3, 5, 7 and 17, with 37838430.
a(6) = 900569670 as 900569670 = 2 * 3 * 5 * 7 * 11 * 13 * 29989 while sopfr(900569670) = 30030 = 2 * 3 * 5 * 7 * 11 * 13, which shares 6 distinct prime factors, 2, 3, 5, 7, 11 and 13, with 900569670.
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CROSSREFS
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KEYWORD
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nonn,more,new
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AUTHOR
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STATUS
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approved
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0, 2, 0, 2, 0, 4, 2, 2, 0, 2, 0, 4, 2, 3, 2, 2, 0, 2, 0, 2, 0, 6, 4, 4, 2, 2, 2, 3, 2, 3, 2, 2, 0, 2, 0, 2, 0, 4, 2, 2, 0, 2, 0, 6, 4, 4, 4, 4, 2, 2, 2, 2, 2, 5, 3, 3, 2, 2, 2, 3, 2, 3, 2, 2, 0, 2, 0, 2, 0, 4, 2, 2, 0, 2, 0, 4, 2, 3, 2, 2, 0, 2, 0, 2, 0, 8, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2
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OFFSET
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0,2
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LINKS
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PROG
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(PARI)
A050602sq(x, y) = if(!bitand(x, y), 0, 1+A050602sq(bitxor(x, y), 2*bitand(x, y)));
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CROSSREFS
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Cf. A002450 (seems to give the positions of records).
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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6, -48, 2560, -1572864, -3848290697216, 6649092007880460460883968, -18999521285301737936647902825311679255527123058688, 76895533293152762966220781422103876125697362804839499718093497881599910128103059800826635129716736
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OFFSET
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0,1
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COMMENTS
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The difference between A372444(n) and the term of A086893 with the same binary length.
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LINKS
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FORMULA
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EXAMPLE
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The term of A086893 that has same binary length as A372444(0) = 27 is 21 [as 21 = 10101_2 in binary, and 27 = 11011_2 in binary], therefore a(0) = 27-21 = 6.
The term of A086893 that has same binary length as A372444(1) = 165 is 213, therefore a(1) = 165-213 = -48.
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PROG
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(PARI)
A371094(n) = { my(m=1+3*n, e=valuation(m, 2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A086893(n) = (if(n%2, 2^(n+1), 2^(n+1)+2^(n-1))\3);
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CROSSREFS
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KEYWORD
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sign,new
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AUTHOR
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STATUS
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approved
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2, 6, 10, 21, 41, 80, 162, 324, 646, 1294, 2586, 5173, 10345, 20691, 41381, 82760, 165522, 331044, 662089, 1324177, 2648353, 5296707, 10593413, 21186827, 42373652, 84747305, 169494609, 338989216, 677978435, 1355956869, 2711913736, 5423827472, 10847654948, 21695309896, 43390619791, 86781239586, 173562479173, 347124958344
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OFFSET
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0,1
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COMMENTS
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The formula involving A372451 and A372453 shows that each term is at most +-1 from the corresponding term of A372451, that are the first differences of A372449.
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LINKS
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FORMULA
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a(n) = A372451(n) + [A372453(n)<=0] - [A372453(1+n)<0], where [ ] is the Iverson bracket, yielding 1 or 0 depending on whether the given inequivalence holds or does not hold.
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EXAMPLE
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Between A372444(1)=165 and A372444(2)=8021 there are six terms (213, 341, 853, 1365, 3413, 5461) of A086893, therefore a(1) = 6.
Between A372444(2)=8021 and A372444(3)=12408149 there are 10 terms (13653, 21845, 54613, 87381, 218453, 349525, 873813, 1398101, 3495253, 5592405) of A086893, therefore a(2) = 10.
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PROG
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A372230
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Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.
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1
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1, 7, 7, 35, 105, 168, 155, 1085, 5208, 13888, 651, 9765, 109368, 874944, 3999744, 2667, 82677, 1984248, 37039296, 507967488, 4063739904, 10795, 680085, 33732216, 1349288640, 43177236480, 1036253675520, 14737830051840
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OFFSET
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2,2
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COMMENTS
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For n>=2 and 3<=k<=n, to construct a size k circuit of M[A]: Choose a basis b_1,b_2,...,b_{k-1} of a k-1 dimensional subspace of GF(2)^n. Append the vector b_1 + b_2 + ... + b_{k-1}.
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REFERENCES
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J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 1992, page 8.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins ...
1;
7, 7;
35, 105, 168;
155, 1085, 5208, 13888;
651, 9765, 109368, 874944, 3999744;
2667, 82677, 1984248, 37039296, 507967488, 4063739904;
...
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MATHEMATICA
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nn = 8; Map[Select[#, # > 0 &] &, Table[Table[PadRight[Table[Product[(2^n - 2^i)/(2^k - 2^i), {i, 0, k - 1}], {k, 2, n}], nn], {n, 2, nn}][[All, j]]* Table[Product[2^n - 2^i, {i, 0, n - 1}]/(n + 1)!, {n, 2, nn}][[j]], {j, 1, nn - 1}] // Transpose] // Grid
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A372582
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Numbers k such that 2*k + 3 and 3*k + 2 are semiprimes.
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11, 24, 31, 44, 69, 71, 92, 99, 100, 101, 107, 108, 109, 123, 125, 128, 131, 132, 148, 160, 181, 184, 204, 207, 224, 235, 243, 245, 249, 251, 263, 267, 271, 288, 289, 297, 304, 332, 347, 348, 355, 357, 359, 360, 364, 371, 373, 380, 384, 389, 400, 420, 423, 445, 448, 449, 451, 459, 460, 465, 485
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OFFSET
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1,1
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COMMENTS
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The first k where k, k + 1 and k + 2 are all terms is 99. There is no k where k, k + 1, k + 2 and k + 3 are all terms, because 3 * t + 2 is divisible by 4 for t = one of these.
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LINKS
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EXAMPLE
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a(3) = 31 is a term because 2 * 31 + 3 = 65 = 5 * 13 and 3 * 31 + 2 = 95 = 5 * 19 are both semiprimes.
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MAPLE
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filter:= k -> numtheory:-bigomega(3*k+2) = 2 and numtheory:-bigomega(2*k+3) = 2:
select(filter, [$1..1000]);
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MATHEMATICA
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s = {}; Do[If[2 == PrimeOmega[2*k + 3] == PrimeOmega[3*k +
2], AppendTo[s, k]], {k, 10^3}]; s
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A372550
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Primes such that the next 10 prime gaps are all distinct.
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15919, 15923, 24113, 24517, 30509, 34883, 34897, 36107, 49201, 52747, 56249, 64927, 64937, 66107, 66109, 66191, 67247, 67261, 67271, 67273, 68147, 70639, 70657, 70663, 70667, 70687, 70709, 70717, 71549, 75797, 78317, 78929, 79979, 81083, 81101, 83701, 88301, 94117, 94603, 94613, 96497, 97609
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(3) = 24113 is a term because it is prime, the next 10 primes are 24121, 24133, 24137, 24151, 24169, 24179, 24181, 24197, 24203, 24223, and the gaps between these 11 primes are 8, 12, 4, 14, 18, 10, 2, 16, 6, 20 which are all distinct.
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MAPLE
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P:= [seq(ithprime(i), i=1..11)]:
R:= NULL: count:= 0:
while count < 100 do
P:= [op(P[2..-1]), nextprime(P[-1])];
if nops(convert(P[2..-1]-P[1..-2], set))=10 then
count:= count+1; R:= R, P[1];
fi
od:
R;
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MATHEMATICA
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s = {};
Do[If[10 == Length[Union[Differences[Prime[Range[k, k + 10]]]]], AppendTo[s,
Prime[k]]], {k, , 10000}]; s
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A372528
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Expansion of g.f. A(x) satisfying A( -x * A( x - x^2 ) ) = -x^2.
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1, 1, 3, 8, 22, 65, 200, 637, 2090, 7021, 24041, 83611, 294511, 1048376, 3765080, 13623820, 49617990, 181733222, 668947823, 2473277248, 9180700787, 34200489886, 127819746470, 479124333321, 1800838945043, 6785517883825, 25626477179000, 96988079848223, 367794448974300, 1397301289617580
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with its series reversion R(x), satisfy the following formulas.
(1) A( -x*A(x - x^2) ) = -x^2.
(2) A(x - x^2) = R(-x^2)/(-x).
(3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.
(4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).
(5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).
(6) A(x) = -A(x - 2*C(x)) where C(x) = -C(x - 2*C(x)) is a g.f. of the Catalan numbers (A000108).
(7) A( -A(x)*C(x) ) = -C(x)^2 where C(x) = (1 - sqrt(1 - 4*x))/2 is a g.f. of the Catalan numbers (A000108).
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EXAMPLE
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G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 22*x^5 + 65*x^6 + 200*x^7 + 637*x^8 + 2090*x^9 + 7021*x^10 + 24041*x^11 + 83611*x^12 + ...
RELATED SERIES.
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^3 + 2*x^4 + 4*x^5 - 9*x^6 - 18*x^7 + 44*x^8 + 91*x^9 - 234*x^10 - 496*x^11 + 1318*x^12 + ...
where A(x - x^2) = R(-x^2)/(-x).
Also, the bisections B1 and B2 of R(x) are
B1 = (R(x) - R(-x))/2 = x - x^3 + 4*x^5 - 18*x^7 + 91*x^9 - 496*x^11 + 2839*x^13 - 16836*x^15 + 102545*x^17 - 637733*x^19 + ...
B2 = (R(x) + R(-x))/2 = -x^2 + 2*x^4 - 9*x^6 + 44*x^8 - 234*x^10 + 1318*x^12 - 7722*x^14 + 46594*x^16 - 287611*x^18 + 1807720*x^20 + ...
and satisfy B1^2 + B2 = 0 and A(-x*B1) = -B1^2.
SPECIFIC VALUES.
A( -A(2/9) / 3 ) = -1/9 where
A(2/9) = 0.3655811677545134614272600644874552972994602150418984...
A( -A(3/16) / 4 ) = -1/16 where
A(3/16) = 0.2645434685642398513217156896362957133168212272114320...
A( -A(4/25) / 5 ) = -1/25 where
A(4/25) = 0.2076566162630115730635446744577181791494166261819659...
A( -A(5/36) / 6 ) = -1/36 where
A(5/36) = 0.1711609712404346976409014231532840797963445277760447...
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PROG
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(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( x^2 + subst(Ser(A), x, -x*subst(Ser(A), x, x - x^2) ), #A)); A[n+1]}
for(n=1, 35, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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