A184395 - OEIS

%I #17 Jun 20 2023 16:18:09

%S 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,

%T 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,

%U 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

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

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

%C 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

%H Antti Karttunen, <a href="/A184395/b184395.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 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.

%e From _Antti Karttunen_, Aug 24 2017: (Start)

%e 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.

%e 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.

%e (End)

%t Table[Length[Union[DivisorSigma[1,Divisors[n]]]],{n,120}] (* _Harvey P. Dale_, Jun 20 2023 *)

%o (PARI) A184395(n) = length(vecsort(apply(d->sigma(d),divisors(n)), , 8)); \\ _Antti Karttunen_, Aug 24 2017

%Y Cf. A000005, A000203, A184396.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Jan 12 2011

%E Name changed, a(66) and a(70) corrected and more terms added by _Antti Karttunen_, Aug 24 2017