A320003 - OEIS

%I #20 Nov 25 2023 07:59:50

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

%T 0,2,0,0,1,0,0,2,0,0,3,0,0,1,0,0,1,0,0,3,0,0,1,0,0,2,0,0,3,0,0,2,0,0,

%U 1,0,0,2,0,0,2,0,0,2,0,0,3,0,0,2,0,0,1,0,0,4,0,0,1,0,0,1,0,0,3,0,0,2,0,0,3

%N Number of proper divisors of n of the form 6*k + 3.

%C Number of divisors of n that are odd multiples of 3 and less than n.

%H Antti Karttunen, <a href="/A320003/b320003.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) = Sum_{d|n, d<n} (1-A000035(d))*A079978(d).

%F a(n) = A007814(A319990(n)).

%F a(4*n) = a(2*n). - _David A. Corneth_, Oct 03 2018

%F G.f.: Sum_{k>=1} x^(12*k-6) / (1 - x^(6*k-3)). - _Ilya Gutkovskiy_, Apr 14 2021

%F Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,6) - (2 - gamma)/6 = -0.208505..., gamma(3,6) = -(psi(1/2) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023

%e For n = 18, of its five proper divisors [1, 2, 3, 6, 9] only 3 and 9 are odd multiples of three, thus a(18) = 2.

%e For n = 108, the odd part is 27 for which 27/3 has 3 divisors. As 108 is even, we don't subtract 1 from that 3 to get a(108) = 3. - _David A. Corneth_, Oct 03 2018

%t a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 3 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)

%o (PARI) A320003(n) = if(!n,n,sumdiv(n, d, (d<n)*(3==(d%6))));

%o (PARI) a(n) = if(n%3==0, my(v=valuation(n, 2)); n>>=v; numdiv(n/3)-(!v), 0) \\ _David A. Corneth_, Oct 03 2018

%Y Cf. A319990, A320001, A320005.

%Y Cf. A001620, A016629, A020759 (psi(1/2)).

%K nonn,easy

%O 1,18

%A _Antti Karttunen_, Oct 03 2018