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

1,5

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

Links

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Index entries for sequences related to partitions

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.

Example

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.

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}]

Prog

(PARI) a(n) = n + 1 - numdiv(2*n); \\ Michel Marcus, Dec 23 2013
(GAP) List([1..10^4], n->n+1-Tau(2*n)); # Muniru A Asiru, Feb 04 2018

Crossrefs

Cf. A000005, A066660.

Sequence in context: A136692 A368673 A219194 * A223012 A101452 A019963

Adjacent sequences: A234303 A234304 A234305 * A234307 A234308 A234309

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 22 2013

Status

approved