login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Number of proper divisors of n that are either of the form 6*k+1 or 6*k + 5.
5

%I #15 Nov 25 2023 08:00:35

%S 0,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,2,2,2,1,1,2,2,1,2,1,2,1,1,2,2,

%T 3,1,1,2,2,2,1,2,1,2,2,2,1,1,2,3,2,2,1,1,3,2,2,2,1,2,1,2,2,1,3,2,1,2,

%U 2,4,1,1,1,2,3,2,3,2,1,2,1,2,1,2,3,2,2,2,1,2,3,2,2,2,3,1,1,3,2,3,1,2,1,2,4

%N Number of proper divisors of n that are either of the form 6*k+1 or 6*k + 5.

%H Antti Karttunen, <a href="/A320015/b320015.txt">Table of n, a(n) for n = 1..65537</a>

%H R. A. Smith and M. V. Subbarao, <a href="https://doi.org/10.4153/CMB-1981-005-3">The average number of divisors in an arithmetic progression</a>, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

%F a(n) = A320001(n) + A320005(n).

%F a(n) = A035218(n) - ch15(n), where ch15 is the characteristic function of numbers of the form +-1 mod 6, i.e., ch15(n) = A232991(n-1).

%F Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = 2*(gamma + log(12)/4 - 1)/3 = 0.132294..., and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023

%t a[n_] := DivisorSum[n, 1 &, # < n && MemberQ[{1, 5}, Mod[#, 6]] &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)

%o (PARI) A320015(n) = if(!n,n,sumdiv(n, d, (d<n)&&(d%2)&&(d%3)));

%Y Cf. A001620, A035218, A232991, A293895, A293896, A320001, A320005.

%K nonn,easy

%O 1,10

%A _Antti Karttunen_, Oct 03 2018