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A343525
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If n = Product (p_j^k_j) then a(n) = Product (2*p_j^k_j + 1), with a(1) = 1.
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3
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1, 5, 7, 9, 11, 35, 15, 17, 19, 55, 23, 63, 27, 75, 77, 33, 35, 95, 39, 99, 105, 115, 47, 119, 51, 135, 55, 135, 59, 385, 63, 65, 161, 175, 165, 171, 75, 195, 189, 187, 83, 525, 87, 207, 209, 235, 95, 231, 99, 255, 245, 243, 107, 275, 253, 255, 273, 295, 119, 693, 123, 315, 285, 129, 297, 805
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * usigma(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-1) - 2/p^(2*s-1)). - Amiram Eldar, Jul 24 2024
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MAPLE
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a:= n-> mul(2*i[1]^i[2]+1, i=ifactors(n)[2]):
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MATHEMATICA
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a[1] = 1; a[n_] := Times @@ ((2 #[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 66}]
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PROG
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(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1] = 2*f[k, 1]^f[k, 2]+1; f[k, 2]=1); factorback(f); \\ Michel Marcus, Apr 18 2021
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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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