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A074789
Partial sums of usigma(n)^2: square of the sum of unitary divisors of n.
4
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
1,2
LINKS
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.
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
MATHEMATICA
Accumulate[Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &]^2, {n, 1, 50}]] (* Vaclav Kotesovec, Feb 04 2019 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
STATUS
approved