A343569 If n = Product (p_j^k_j) then a(n) = Product (2*(p_j^k_j + 1)), with a(1) = 1.
1, 6, 8, 10, 12, 48, 16, 18, 20, 72, 24, 80, 28, 96, 96, 34, 36, 120, 40, 120, 128, 144, 48, 144, 52, 168, 56, 160, 60, 576, 64, 66, 192, 216, 192, 200, 76, 240, 224, 216, 84, 768, 88, 240, 240, 288, 96, 272, 100, 312, 288, 280, 108, 336, 288, 288, 320, 360, 120, 960, 124, 384, 320, 130
Offset: 1
Programs
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Mathematica
a[1] = 1; a[n_] := Times @@ (2 (#[[1]]^#[[2]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 64}] -
PARI
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2*f[k,1]^f[k,2] + 2; f[k,2] = 1); factorback(f); \\ Michel Marcus, Apr 20 2021
Formula
a(n) = usigma(n) * 2^omega(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d) * usigma(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * 2^omega(d) * 2^omega(n/d).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} A343525(d).