A234306 a(n) = n + 1 - d(2n), where d(n) is the number of divisors of n.

0, 0, 0, 1, 2, 1, 4, 4, 4, 5, 8, 5, 10, 9, 8, 11, 14, 10, 16, 13, 14, 17, 20, 15, 20, 21, 20, 21, 26, 19, 28, 26, 26, 29, 28, 25, 34, 33, 32, 31, 38, 31, 40, 37, 34, 41, 44, 37, 44, 42, 44, 45, 50, 43, 48, 47, 50, 53, 56, 45, 58, 57, 52, 57, 58, 55, 64, 61

Offset: 1

Author

Wesley Ivan Hurt, Dec 22 2013

Keywords

• nonn
• easy

Comments

Number of partitions of 2n into exactly two parts: (2n-i,i) such that i does not divide 2n-i. Complement of A066660.

Number of positive integers k <= n, such that k does not divide 2n-k. For example, a(12) = 5 since there are 5 positive integers k less than or equal to 12 that do not divide 2*12-k. They are 5, 7, 9, 10, and 11. - Wesley Ivan Hurt, Jun 24 2021

Examples

			a(6) = 1; In this case, 2(6) = 12 has exactly 6 partitions into two parts: (11,1), (10,2), (9,3), (8,4), (7,5), (6,6).  Note that 5 does not divide 7 but the smallest parts of the other partitions divide their corresponding largest parts.  Therefore, a(6) = 1.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..5000
• Index entries for sequences related to partitions

Crossrefs

Cf. A000005, A066660.

Programs

GAP

List([1..10^4], n->n+1-Tau(2*n)); # Muniru A Asiru, Feb 04 2018

Maple

with(numtheory); A234306:=n->n + 1 - tau(2*n); seq(A234306(n), n=1..100);

Mathematica

Table[n + 1 - DivisorSigma[0, 2n], {n, 100}]

PARI

a(n) = n + 1 - numdiv(2*n); \\ Michel Marcus, Dec 23 2013

Formula

a(n) = n + 1 - A000005(2n).

a(n) = n - A066660(n).

a(n) = Sum_{i=1..n | i does not divide 2n-i} 1.