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 A369758 The sum of divisors of the smallest cubefull exponentially odd number that is divisible by n. 3
 1, 15, 40, 15, 156, 600, 400, 15, 40, 2340, 1464, 600, 2380, 6000, 6240, 63, 5220, 600, 7240, 2340, 16000, 21960, 12720, 600, 156, 35700, 40, 6000, 25260, 93600, 30784, 63, 58560, 78300, 62400, 600, 52060, 108600, 95200, 2340, 70644, 240000, 81400, 21960, 6240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS First differs from A369720 at n = 16. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 FORMULA a(n) = A000203(A356192(n)). Multiplicative with a(p) = p^3 + p^2 + p + 1, a(p^e) = (p^(e+1)-1)/(p-1) for an odd e >= 3, and a(p^e) = (p^(e+2)-1)/(p-1) for an even e. a(n) >= A000203(n), with equality if and only if n is cubefull exponentially odd number (A335988). Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-5) - 1/p^(3*s-4) - 1/p^(3*s-3) + 1/p^(4*s-5) + 1/p^(4*s-4)). Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * zeta(6) * Product_{p prime} (1 - 1/p^4 - 1/p^6 + 1/p^10 + 1/p^11 - 1/p^13) = 1.00040193512214077945... . Equivalently, c = Product_{p prime} (1 + 1/(p^3*(p^4 - 1)*(p^4 + p^2 + 1))). - Vaclav Kotesovec, Feb 02 2024 MATHEMATICA f[p_, e_] := (p^If[OddQ[e], Max[e, 3] + 1, e + 2] - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] PROG (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^if(f[i, 2]%2, max(f[i, 2], 3) + 1, f[i, 2] + 2) - 1)/(f[i, 1] - 1)); } (Python) from math import prod from sympy import factorint def A369758(n): return prod((p**((3 if e==1 else e)+1+(e&1^1))-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Feb 03 2024 CROSSREFS Cf. A000203, A335988, A356192, A365349, A369720, A369757, A369759. Cf. A013662, A013664. Sequence in context: A091847 A062222 A369720 * A325659 A223432 A044092 Adjacent sequences: A369755 A369756 A369757 * A369759 A369760 A369761 KEYWORD nonn,easy,mult AUTHOR Amiram Eldar, Jan 31 2024 STATUS approved

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Last modified June 23 01:26 EDT 2024. Contains 373629 sequences. (Running on oeis4.)