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

1,3

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).

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

Mathematica

Table[Length[Select[Range[n], Divisible[n, DivisorSigma[1, #]]&]], {n, 1, 100}] (* Vaclav Kotesovec, Feb 16 2019 *)

Prog

(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).

Crossrefs

Cf. A000203, A070610.

Row lengths of A378912.

Sequence in context: A241605 A128180 A209279 * A322529 A349526 A329949

Adjacent sequences: A074751 A074752 A074753 * A074755 A074756 A074757

Keyword

nonn

Author

Benoit Cloitre, Sep 28 2002

Status

approved