A378450 a(n) is the number of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

0, 0, 1, 0, 3, 0, 5, 0, 6, 3, 9, 0, 11, 9, 9, 0, 15, 0, 17, 0, 17, 21, 21, 0, 24, 27, 25, 0, 27, 0, 29, 0, 33, 39, 33, 0, 35, 45, 41, 0, 39, 0, 41, 21, 23, 57, 45, 0, 50, 30, 57, 35, 51, 0, 57, 0, 65, 75, 57, 0, 59, 81, 45, 0, 69, 0, 65, 63, 81, 2, 69, 0, 71, 99, 61, 77, 81, 0, 77, 0, 90, 111, 81, 0, 93, 117, 105

Offset: 1

Author

Antti Karttunen, Nov 29 2024

Keywords

• nonn

Comments

Difference between the sum of divisors n and the number of distinct sums of distinct divisors of n.

Examples

			For n = 3, with divisors [1, 3] and sigma(3)=4, only 2 in range 1..4 cannot be represented as a sum of a subset of [1, 3], therefore a(3) = 1.
For n = 15, with divisors [1, 3, 5, 15] and sigma(15) = 24, the subset sums are 1, 3, 1+3, 5, 1+5, 3+5, 1+3+5, 15, 1+15, 3+15, 1+3+15, 5+15, 1+5+15, 3+5+15, 1+3+5+15 i.e., [1, 3, 4, 5, 6, 8, 9, 15, 16, 18, 19, 20, 21, 23, 24], which leaves 2, 7, 10, 11, 12, 13, 14, 17, 22 as unrepresented numbers, therefore a(15) = 9.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000203, A119347, A237289 (gives the sums of unrepresented numbers), A322860.

Cf. A005153 (positions of 0's), A237287 (of nonzeros), A030057.

Programs

PARI

A119347(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; };
A378450(n) = (sigma(n)-A119347(n));

Formula

a(n) = A000203(n) - A119347(n).