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Sum of proper divisors of n of the form 3k+1.
6

%I #17 Nov 27 2023 03:04:24

%S 0,1,1,1,1,1,1,5,1,1,1,5,1,8,1,5,1,1,1,15,8,1,1,5,1,14,1,12,1,11,1,21,

%T 1,1,8,5,1,20,14,15,1,8,1,27,1,1,1,21,8,36,1,18,1,1,1,40,20,1,1,15,1,

%U 32,8,21,14,23,1,39,1,18,1,5,1,38,26,24,8,14,1,71,1,1,1,40,1,44,1,27,1,11,21,51,32,1,20,21,1,57,1,40,1,35,1,70,8

%N Sum of proper divisors of n of the form 3k+1.

%H Antti Karttunen, <a href="/A293897/b293897.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.

%F a(n) = A078181(n) - ([n == 1 (mod 3)]*n).

%F G.f.: Sum_{k>=1} (3*k-2) * x^(6*k-4) / (1 - x^(3*k-2)). - _Ilya Gutkovskiy_, Apr 14 2021

%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/36 - 1/6 = 0.107489... . - _Amiram Eldar_, Nov 27 2023

%t Table[DivisorSum[n, # &, And[Mod[#, 3] == 1, # != n] &], {n, 105}] (* _Michael De Vlieger_, Nov 08 2017 *)

%o (PARI) A293897(n) = sumdiv(n,d,(d<n)*(1==(d%3))*d);

%Y Cf. A078181, A293901, A293895, A293898.

%K nonn,easy

%O 1,8

%A _Antti Karttunen_, Nov 06 2017