A161116 a(n) is the number of nontrivial positive divisors of 2n+3.

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 2, 0, 0, 4, 0, 1, 2, 0, 2, 2, 0, 0, 4, 2, 0, 2, 0, 0, 4, 2, 0, 3, 0, 2, 2, 0, 2, 2, 2, 0, 4, 0, 0, 6, 0, 0, 2, 0, 2, 4, 2, 1, 2, 2, 0, 2, 0, 2, 6, 0, 0, 2, 2, 2, 4, 0, 0, 4, 2, 0, 2, 2, 0, 6, 0, 1, 4, 0, 4, 2, 0, 0, 2

Offset

0,7

Comments

a(n) = 0 iff n is in A067076, i.e., 2n+3 is prime.

a(n) is the number of positive integers of the form (n-3k)/(2k+1), 1 <= k <= n/3.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Formula

a(n) = A070824(2n+3).

For n >= 1, a(n) = A160973(n) + A079978(n). - Vladimir Shevelev, Jun 07 2009

Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma + 3*log(2) - 5)*n/2, where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 10 2026

Example

Since for n = 3 we have 2n+3 = 9 and only nontrivial divisor of 9 is 3, then a(3) = 1.

Mathematica

a[n_] := DivisorSigma[0, 2*n+3] - 2; Array[a, 100, 0] (* Amiram Eldar, Mar 10 2026 *)

Prog

(PARI) a(n) = numdiv(2*n+3) - 2; \\ Michel Marcus, Feb 08 2016
(Magma) [NumberOfDivisors(2*n+3)-2: n in [0..100]]; // Vincenzo Librandi, Feb 08 2016

Crossrefs

Cf. A001620, A006254, A034953, A054269, A067076, A070824, A079978, A082749, A160973.

Sequence in context: A118683 A175800 A344981 * A262726 A112605 A111775

Adjacent sequences: A161113 A161114 A161115 * A161117 A161118 A161119

Keyword

nonn,easy

Author

Vladimir Shevelev, Jun 02 2009

Extensions

Edited by Charles R Greathouse IV, Oct 12 2009

More terms from Michel Marcus, Feb 08 2016

Status

approved