A066728 a(n) is the number of integers of the form (n+k+n*k)/(n-k) for k = 1,2,...,n-1.

0, 1, 1, 3, 1, 4, 2, 4, 2, 7, 1, 7, 3, 5, 3, 8, 1, 11, 3, 7, 3, 9, 2, 9, 5, 7, 3, 15, 1, 13, 3, 6, 7, 11, 3, 11, 3, 9, 3, 19, 1, 15, 5, 7, 5, 11, 2, 17, 5, 11, 3, 15, 3, 19, 7, 9, 3, 15, 1, 15, 5, 7, 11, 15, 3, 15, 3, 15, 3, 29, 1, 14, 5, 7, 11, 15, 3, 23, 4, 11, 4, 15, 3, 15, 7, 9, 3, 29, 3, 23

Offset: 1

Author

Benoit Cloitre, Jan 15 2002

Keywords

• nonn
• easy

Comments

a(n) = 1 iff n is 2 or the lesser of twin primes (for n >= 3, n follows the sequence A001359).

Also the number of factors of n*(n+2) which are less than n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 02 2003

Examples

			(4 + 1 + 4*1)/(4 - 1), (4 + 2 + 4*2)/(4 - 2), and (4 + 3 + 4*3)/(4 - 1) are integers, hence a(4) = 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001359, A063091.

Programs

Maple

with(numtheory):A066728 := n->ceil(tau(n*(n+2))/2)-1;

Mathematica

a[n_] := Ceiling[DivisorSigma[0, n*(n+2)] / 2]  - 1; Array[a, 100] (* Amiram Eldar, Feb 01 2025 *)

PARI

a(n) = ceil(numdiv(n*(n+2))/2) - 1; \\ Amiram Eldar, Feb 01 2025

Formula

a(n) = ceiling( d(n*(n+2)) / 2 ) - 1, where d(n) = number of divisors of n (A000005). - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 02 2003