A095374 - OEIS

%I #35 Dec 20 2022 01:40:29

%S 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,

%T 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,

%U 1,3,3,3,5,1,1,5,3,1,3,3,1,7,1,2,5,1,5

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

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

%H Michael De Vlieger, <a href="/A095374/b095374.txt">Table of n, a(n) for n = 1..5000</a>

%H Gerzson Kéri, <a href="http://ac.inf.elte.hu/Vol_053_2022/093_53.pdf">The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas</a>, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.

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

%F 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

%F 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

%F a(n) = A099774(n+1) - 1. - _Bernard Schott_, Mar 04 2019

%e A095372(22) is divisible by {91, 9091, 90909091, 90909090909091, A095372(22)}, thus a(22)=5.

%e 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 + ...

%t 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}]

%t Table[DivisorSigma[0,2n+1],{n,90}]-1 (* _Harvey P. Dale_, Oct 31 2015 *)

%t Table[Sum[1 - Ceiling@ # + Floor@ # &[(n + i)/(n - i + 1)], {i, n}], {n, 87}] (* _Michael De Vlieger_, Feb 27 2017 *)

%o (PARI) {a(n) = if( n<0, 0, numdiv(2*n + 1) - 1)} /* _Michael Somos_, Aug 30 2012 */

%Y Cf. A000005, A095372, A095373, A023645, A099774.

%K nonn,easy

%O 1,4

%A _Labos Elemer_, Jun 07 2004