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

%I A074789 #22 Jul 24 2024 18:12:03
%S A074789 1,10,26,51,87,231,295,376,476,800,944,1344,1540,2116,2692,2981,3305,
%T A074789 4205,4605,5505,6529,7825,8401,9697,10373,12137,12921,14521,15421,
%U A074789 20605,21629,22718,25022,27938,30242,32742,34186,37786,40922,43838
%N A074789 Partial sums of usigma(n)^2: square of the sum of unitary divisors of n.
%H A074789 Vaclav Kotesovec, <a href="/A074789/b074789.txt">Table of n, a(n) for n = 1..10000</a>
%H A074789 Vaclav Kotesovec, <a href="/A074789/a074789.jpg">Graph - the asymptotic ratio</a>.
%H A074789 László Tóth, An asymptotic formula concerning the unitary divisor sum function, Stud. Univ. Babes-Bolyai, Math., Vol. 34, No. 2 (1989), pp. 3-10; <a href="https://www.cs.ubbcluj.ro/~studia-m/old_issues/subbmath_1989_34_02.pdf">entire volume</a>.
%F A074789 a(n) = Sum_{k=1..n} usigma(k)^2 = Sum_{k=1..n} A034448(k)^2.
%F A074789 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.
%F A074789 alpha_1 = 1.001619936509160661474009830789... . - _Amiram Eldar_, Jul 24 2024
%t A074789 Accumulate[Table[DivisorSum[n, # &, CoprimeQ[#, n/#] &]^2, {n, 1, 50}]] (* _Vaclav Kotesovec_, Feb 04 2019 *)
%o A074789 (PARI) A034448(n) = {my(f = factor(n)); prod(i=1, #f~, 1 + f[i, 1]^f[i, 2]);}
%o A074789 lista(nmax) = {my(s = 0); for(n = 1, nmax, s += A034448(n)^2; print1(s, ", "));} \\ _Amiram Eldar_, Jul 24 2024
%Y A074789 Cf. A034448, A064609.
%Y A074789 Cf. A057434, A061502, A072379.
%Y A074789 Cf. A002117, A013661.
%K A074789 nonn,easy
%O A074789 1,2
%A A074789 Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002