A095374 One less than the number of divisors of 2*n + 1.

1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 3, 3, 1, 3, 1, 1, 5, 1, 2, 3, 1, 3, 3, 1, 1, 5, 3, 1, 3, 1, 1, 5, 3, 1, 4, 1, 3, 3, 1, 3, 3, 3, 1, 5, 1, 1, 7, 1, 1, 3, 1, 3, 5, 3, 2, 3, 3, 1, 3, 1, 3, 7, 1, 1, 3, 3, 3, 5, 1, 1, 5, 3, 1, 3, 3, 1, 7, 1, 2, 5, 1, 5

Offset: 1

Author

Labos Elemer, Jun 07 2004

Keywords

• nonn
• easy

Comments

Number of special divisors of A095372(n) with A095372(k) form.

Examples

			A095372(22) is divisible by {91, 9091, 90909091, 90909090909091, A095372(22)}, thus a(22)=5.
G.f.= x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 3*x^7 + x^8 + x^9 + 3*x^10 + x^11 + 2*x^12 + ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..5000
• Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

Crossrefs

Cf. A000005, A001620, A095372, A095373, A023645, A099774.

Programs

Mathematica

g[x_]:=1+90*(100^x-1)/99 t=Table[1+90*(100^n-1)/99, {n, 1, 35}]; Do[Print[{w, is=Intersection[Divisors[g[w]], t], Length[is]}], {w, 1, 35}]
Table[DivisorSigma[0,2n+1],{n,90}]-1 (* Harvey P. Dale, Oct 31 2015 *)
Table[Sum[1 - Ceiling@ # + Floor@ # &[(n + i)/(n - i + 1)], {i, n}], {n, 87}] (* Michael De Vlieger, Feb 27 2017 *)

PARI

{a(n) = if( n<0, 0, numdiv(2*n + 1) - 1)} /* Michael Somos, Aug 30 2012 */

Formula

a(n) = A023645(2*n + 1) = A000005(2*n + 1) - 1.

a(n) = Sum_{i=1..n} 1-ceiling((n+i)/(n-i+1))+floor((n+i)/(n-i+1)). - Wesley Ivan Hurt, Feb 26 2017

O.g.f.: Sum_{n >= 1} x^n/(1 - x^(2*n+1)) = x/(1 - x) + Sum_{n >= 1} x^(2*n*(n+1))*(1 + x^(2*n+1))/(1 - x^(2*n+1)). - Peter Bala, Mar 04 2019

a(n) = A099774(n+1) - 1. - Bernard Schott, Mar 04 2019

Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3 + 3*log(2)) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 15 2025