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A325240
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Numbers whose minimum prime exponent is 2.
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5
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4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800
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OFFSET
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1,1
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COMMENTS
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Or barely powerful numbers, a subset of powerful numbers A001694.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023
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EXAMPLE
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The sequence of terms together with their prime indices begins:
4: {1,1}
9: {2,2}
25: {3,3}
36: {1,1,2,2}
49: {4,4}
72: {1,1,1,2,2}
100: {1,1,3,3}
108: {1,1,2,2,2}
121: {5,5}
144: {1,1,1,1,2,2}
169: {6,6}
196: {1,1,4,4}
200: {1,1,1,3,3}
225: {2,2,3,3}
288: {1,1,1,1,1,2,2}
289: {7,7}
324: {1,1,2,2,2,2}
361: {8,8}
392: {1,1,1,4,4}
400: {1,1,1,1,3,3}
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MATHEMATICA
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Select[Range[1000], Min@@FactorInteger[#][[All, 2]]==2&]
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PROG
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(PARI) is(n)={my(e=factor(n)[, 2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023
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CROSSREFS
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Maximum instead of minimum gives A067259.
Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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