A074754 Number of integers k such that sigma(k) divides n.
1, 1, 2, 2, 1, 3, 2, 3, 2, 1, 1, 6, 2, 3, 3, 3, 1, 5, 1, 3, 3, 1, 1, 10, 1, 2, 2, 5, 1, 5, 3, 5, 2, 1, 2, 9, 1, 2, 4, 5, 1, 8, 1, 3, 3, 1, 1, 13, 2, 1, 2, 3, 1, 7, 1, 8, 3, 1, 1, 12, 1, 4, 4, 5, 2, 3, 1, 3, 2, 3, 1, 18, 1, 2, 3, 3, 2, 6, 1, 7, 2, 1, 1, 15, 1, 1, 2, 4, 1, 10, 4, 2, 5, 1, 1, 19, 1, 5, 2, 3, 1
Offset: 1
Keywords
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
- Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
- Amiram Eldar, Plot of Sum_{k=1..n} a(k)/(n*log(n)) for n = 10^(1..7).
- Index entries for sequences related to sigma(n).
Programs
-
Mathematica
Table[Length[Select[Range[n], Divisible[n, DivisorSigma[1,#]]&]], {n, 1, 100}] (* Vaclav Kotesovec, Feb 16 2019 *) -
PARI
a(n)=sum(i=1,n,if(n%sigma(i),0,1))
-
PARI
a(n)=if(n<1,0,polcoeff(sum(k=1,n,1/(1-x^sigma(k)),x*O(x^sigma(n))),n))
-
PARI
a(n) = {my(s = []); fordiv(n, d, s = setunion(s, invsigma(d))); #s;} \\ Amiram Eldar, Apr 18 2025, using Max Alekseyev's invphi.gp (see links).
Formula
Sum_{k=1..n} a(k) seems to be asymptotic to c*n*log(n) with c = 0.7...
G.f.: sum(k>=1, 1/(1-x^sigma(k))).
a(n) = Sum_{k=1..n} (1 - ceiling(n/sigma(k)) + floor(n/sigma(k))). - Wesley Ivan Hurt, Apr 21 2023