%I #35 Nov 25 2023 04:33:58
%S 1,1,1,1,2,1,1,1,2,2,1,1,2,1,2,1,2,2,1,2,2,1,1,1,3,2,2,1,2,2,1,1,2,2,
%T 2,2,2,1,2,2,2,2,1,1,4,1,1,1,2,3,2,2,2,2,2,1,2,2,1,2,2,1,3,1,4,2,1,2,
%U 2,2,1,2,2,2,3,1,2,2,1,2,3,2,1,2,4,1,2,1,2,4,2,1,2,1,2,1,2,2,3,3,2,2,1,2,4
%N Number of divisors of n of the form 4k+1.
%C Not multiplicative: a(21) <> a(3)*a(7), for example. - _R. J. Mathar_, Sep 15 2015
%H Nick Hobson, <a href="/A001826/b001826.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</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 G.f.: Sum_{n>0} x^n/(1-x^(4n)) = Sum_{n>=0} x^(4n+1)/(1-x^(4n+1)).
%F a(n) = A001227(n) - A001842(n). - _Reinhard Zumkeller_, Apr 18 2006
%F Sum_{k=1..n} a(k) = n*log(n)/4 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,4) - (1 - gamma)/4 = A256778 - (1 - A001620)/4 = 0.604593... (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%p d:=proc(r,m,n) local i,t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; # no. of divisors i of n with i == r mod m
%p A001826 := proc(n)
%p add(`if`(modp(d,4)=1,1,0),d=numtheory[divisors](n)) ;
%p end proc: # _R. J. Mathar_, Sep 15 2015
%t a[n_] := Count[Divisors[n], d_ /; Mod[d, 4] == 1]; Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, Nov 26 2013 *)
%t a[n_] := DivisorSum[n, 1 &, Mod[#, 4] == 1 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)
%o (PARI) a(n)=if(n<1,0,sumdiv(n,d,d%4==1))
%Y Cf. A001227, A001620, A001842, A256778.
%K nonn,easy
%O 1,5
%A _N. J. A. Sloane_
%E Better definition from _Michael Somos_, Apr 26 2004