A001826 Number of divisors of n of the form 4k+1.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Not multiplicative: a(21) <> a(3)*a(7), for example. - R. J. Mathar, Sep 15 2015

References

• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 132.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 244.

Links

• Nick Hobson, Table of n, a(n) for n = 1..10000
• Michael Gilleland, Some Self-Similar Integer Sequences.
• R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Crossrefs

Cf. A001227, A001620, A001842, A256778.

Programs

Maple

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
A001826 := proc(n)
    add(`if`(modp(d,4)=1,1,0),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Sep 15 2015

Mathematica

a[n_] := Count[Divisors[n], d_ /; Mod[d, 4] == 1]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 26 2013 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 4] == 1 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

PARI

a(n)=if(n<1,0,sumdiv(n,d,d%4==1))

Formula

G.f.: Sum_{n>0} x^n/(1-x^(4n)) = Sum_{n>=0} x^(4n+1)/(1-x^(4n+1)).

a(n) = A001227(n) - A001842(n). - Reinhard Zumkeller, Apr 18 2006

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

Extensions

Better definition from Michael Somos, Apr 26 2004