A184395 a(n) = number of distinct values obtained when sigma is applied to the divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 7, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 8, 8, 4, 2, 12, 2, 8, 4, 10, 2, 8, 4

Offset

1,2

Comments

Sequence is not the same as A000005(n): a(66) = 7, A000005(66) = 8.

a(n) = number of numbers k <= sigma(n) such that k = sigma(d) for some divisor d of n, where sigma = A000203. - This is the original name of the sequence, except that I substituted "some divisor" for "any divisor". - Antti Karttunen, Aug 24 2017

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000203(n) - A184396(n).

Example

For n = 4, sigma(4) = 7, from numbers 1 - 7 there are three numbers k such that k = sigma(d) for any divisor d of n: 1 = sigma(1), 3 = sigma(2), 7 = sigma(4); a(4) = 3.
From Antti Karttunen, Aug 24 2017: (Start)
For n = 66, its 8 divisors are [1, 2, 3, 6, 11, 22, 33, 66]. When applying sigma to these, we obtain [1, 3, 4, 12, 12, 36, 48, 144], with one duplicate present, thus there are only 8-1 = 7 distinct values and a(66) = 7.
For n = 70, its 8 divisors are [1, 2, 5, 7, 10, 14, 35, 70]. When applying sigma to these, we obtain [1, 3, 6, 8, 18, 24, 48, 144], which are all unique values, thus a(70) = 8.
(End)

Mathematica

Table[Length[Union[DivisorSigma[1, Divisors[n]]]], {n, 120}] (* Harvey P. Dale, Jun 20 2023 *)

Prog

(PARI) A184395(n) = length(vecsort(apply(d->sigma(d), divisors(n)), , 8)); \\ Antti Karttunen, Aug 24 2017

Crossrefs

Cf. A000005, A000203, A184396.

Sequence in context: A335519 A167447 A134687 * A329484 A179941 A179942

Adjacent sequences: A184392 A184393 A184394 * A184396 A184397 A184398

Keyword

nonn

Author

Jaroslav Krizek, Jan 12 2011

Extensions

Name changed, a(66) and a(70) corrected and more terms added by Antti Karttunen, Aug 24 2017

Status

approved