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A000113
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Number of transformation groups of order n.
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1
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1, 3, 4, 3, 6, 12, 8, 6, 4, 18, 12, 12, 14, 24, 24, 6, 18, 12, 20, 18, 32, 36, 24, 24, 30, 42, 12, 24, 30, 72, 32, 12, 48, 54, 48, 12, 38, 60, 56, 36, 42, 96, 44, 36, 24, 72, 48, 24, 56, 90, 72, 42, 54, 36, 72, 48, 80, 90, 60, 72, 62, 96, 32, 12, 84, 144, 68
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OFFSET
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1,2
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COMMENTS
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A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 03 2017
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REFERENCES
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B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 139.
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LINKS
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FORMULA
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Let psi(m) = A001615(m) (Dedekind's psi function). Write n = 2^i*3^j*k, where (6,k) = 1 and let i' = floor(i/2) for i < 6, i' = 3 for i >= 6; let j' = 0 for j = 0 or 1, j' = 1 for j >= 2. Then a(n) = psi(n/(2^i'*3^j')) = psi(n)/(2^i'*3^j').
Multiplicative with a(2^e)=3*2^Floor[(e-1)/2] for 0<e<7, a(2^e)=3*2^(e-4) for e>=7, a(3^e)=4 for 0<e<3, a(3^e)=4*3^(e-2) for e>=3 and a(p^e)=(p+1)*p^(e-1) for p>3. - T. D. Noe, Nov 14 2006
Sum_{k=1..n} a(k) ~ c * n^2, where c = 43501/(7680*Pi^2) = 0.573902... . - Amiram Eldar, Oct 23 2022
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EXAMPLE
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G.f. = x + 3*x^2 + 4*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + ...
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MATHEMATICA
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psi[n_] := n*DivisorSum[n, MoebiusMu[#]^2/#&]; a[n_] := (i=IntegerExponent[ n, 2]; j=IntegerExponent[n, 3]; ip = If[i<6, Floor[i/2], 3]; jp = If[j<2, 0, 1]; psi[n]/(2^ip*3^jp)); Array[a, 60] (* Jean-François Alcover, Feb 04 2016 *)
a[ n_] := If[ n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors @ n}] / (2^Min[3, Quotient[IntegerExponent[n, 2], 2]] 3^Boole[1 < IntegerExponent[n, 3]]) ]; (* Michael Somos, Jan 03 2017 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, moebius(d)^2 / d) / (2^min(3, valuation(n, 2)\2) * 3^(1 < valuation(n, 3))))}; /* Michael Somos, Jan 03 2017 */
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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