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A234306
a(n) = n + 1 - d(2n), where d(n) is the number of divisors of n.
3
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
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
Sequence in context: A136692 A368673 A219194 * A223012 A101452 A019963
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Dec 22 2013
STATUS
approved