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

1,5

Comments

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

Links

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

Formula

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

Example

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.

Prog

(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));

Crossrefs

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

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

Sequence in context: A175919 A086664 A164736 * A349343 A109753 A318517

Adjacent sequences: A378447 A378448 A378449 * A378451 A378452 A378453

Keyword

nonn

Author

Antti Karttunen, Nov 29 2024

Status

approved