A203556 a(n) = sigma(n^5).

1, 63, 364, 2047, 3906, 22932, 19608, 65535, 88573, 246078, 177156, 745108, 402234, 1235304, 1421784, 2097151, 1508598, 5580099, 2613660, 7995582, 7137312, 11160828, 6728904, 23854740, 12207031, 25340742, 21523360, 40137576, 21243690, 89572392, 29583456, 67108863

Offset: 1

Author

Paul D. Hanna, Jan 03 2012

Keywords

• nonn
• easy
• mult

Comments

a(n) modulo 6 begins: [1,3,4,1,0,0,0,3,1,0,0,4,0,0,0,1,0,3,0,0,0,0,0,0,1,0,...], in which positions of nonzero residues seem related to squares.

Examples

			L.g.f.: L(x) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...
where the g.f. of A203557 begins:
exp(L(x)) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A203557 (exp), A000203 (sigma), A000584, A013664.

Variants: A065764, A175926, A202994.

Programs

Mathematica

f[p_, e_] := (p^(5*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 09 2020 *)
DivisorSigma[1,Range[40]^5] (* Harvey P. Dale, Dec 05 2021 *)

PARI

a(n) = sigma(n^5)

Formula

Logarithmic derivative of A203557.

Multiplicative with a(p^e) = (p^(5*e+1)-1)/(p-1) for prime p. - Andrew Howroyd, Jul 23 2018

From Amiram Eldar, Nov 05 2022: (Start)

a(n) = A000203(A000584(n)) = A000203(n^5).

Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3220880186... . (End)

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018