A074789 Partial sums of usigma(n)^2: square of the sum of unitary divisors of n.

1, 10, 26, 51, 87, 231, 295, 376, 476, 800, 944, 1344, 1540, 2116, 2692, 2981, 3305, 4205, 4605, 5505, 6529, 7825, 8401, 9697, 10373, 12137, 12921, 14521, 15421, 20605, 21629, 22718, 25022, 27938, 30242, 32742, 34186, 37786, 40922, 43838

Offset: 1

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

Keywords

• nonn
• easy

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio.
• László Tóth, An asymptotic formula concerning the unitary divisor sum function, Stud. Univ. Babes-Bolyai, Math., Vol. 34, No. 2 (1989), pp. 3-10; entire volume.

Crossrefs

Cf. A034448, A064609.

Cf. A057434, A061502, A072379.

Cf. A002117, A013661.

Programs

Mathematica

Accumulate[Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &]^2, {n, 1, 50}]] (* Vaclav Kotesovec, Feb 04 2019 *)

PARI

A034448(n) = {my(f = factor(n)); prod(i=1, #f~, 1 + f[i, 1]^f[i, 2]);}
lista(nmax) = {my(s = 0); for(n = 1, nmax, s += A034448(n)^2; print1(s, ", "));} \\ Amiram Eldar, Jul 24 2024

Formula

a(n) = Sum_{k=1..n} usigma(k)^2 = Sum_{k=1..n} A034448(k)^2.

Asymptotic expression: a(n) = Sum_{k<=n} usigma(k)^2 = (zeta(2)*zeta(3)*alpha_1/3)*n^3 + O(n^2*log(n)^4), alpha_1 = Product_{p prime} (1+1/p^2-2/p^3-1/p^4-2/p^5+3/p^6), zeta(2) = A013661 and zeta(3) = A002117.

alpha_1 = 1.001619936509160661474009830789... . - Amiram Eldar, Jul 24 2024