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Numerator of the quotient sigma(7*n)/sigma(n).
4

%I #14 Mar 22 2024 06:49:13

%S 8,8,8,8,8,8,57,8,8,8,8,8,8,57,8,8,8,8,8,8,57,8,8,8,8,8,8,57,8,8,8,8,

%T 8,8,57,8,8,8,8,8,8,57,8,8,8,8,8,8,400,8,8,8,8,8,8,57,8,8,8,8,8,8,57,

%U 8,8,8,8,8,8,57,8,8,8,8,8,8,57,8,8,8,8,8,8,57,8,8,8,8,8,8,57,8,8,8,8,8,8

%N Numerator of the quotient sigma(7*n)/sigma(n).

%H Amiram Eldar, <a href="/A088841/b088841.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Mar 22 2024: (Start)

%F a(n) = numerator(A283078(n)/A000203(n)).

%F a(n) = (7^(A214411(n)+2)-1)/6 = (49*A268354(n)-1)/6.

%F Sum_{k=1..n} a(k) ~ (7/log(7))*n*log(n) + (9/2 + 7*(gamma-1)/log(7))*n, where gamma is Euler's constant (A001620).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088842(k) = 1 + 36 * Sum_{k>=1} 1/(7^k-1) = 7.87276224676... . (End)

%t Table[Numerator[DivisorSigma[1, 7*n]/DivisorSigma[1, n]], {n, 1, 128}]

%o (PARI) a(n) = numerator(sigma(7*n)/sigma(n)); \\ _Amiram Eldar_, Mar 22 2024

%Y Cf. A088837, A088838, A088839, A088840, A088841, A088842 (denominators).

%Y Cf. A000203, A038712, A080278, A214411, A268354, A283078.

%K nonn,easy,frac

%O 1,1

%A _Labos Elemer_, Nov 04 2003