A001826 - OEIS

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

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OFFSET

1,5

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.

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

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 *)

PROG

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

CROSSREFS

Cf. A001227, A001620, A001842, A256778.

Sequence in context: A317934 A353376 A279848 * A003641 A355241 A165190

Adjacent sequences: A001823 A001824 A001825 * A001827 A001828 A001829