A050469 a(n) = Sum_{ d divides n, n/d=1 mod 4} d - Sum_{ d divides n, n/d=3 mod 4} d.
1, 2, 2, 4, 6, 4, 6, 8, 7, 12, 10, 8, 14, 12, 12, 16, 18, 14, 18, 24, 12, 20, 22, 16, 31, 28, 20, 24, 30, 24, 30, 32, 20, 36, 36, 28, 38, 36, 28, 48, 42, 24, 42, 40, 42, 44, 46, 32, 43, 62, 36, 56, 54, 40, 60, 48, 36, 60, 58, 48, 62, 60, 42, 64, 84, 40
Offset: 1
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
- Index entries for sequences mentioned by Glaisher.
Programs
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Mathematica
max = 70; s = Sum[n*x^(n-1)/(1+x^(2*n)), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 02 2015 *) f[p_, e_] := Which[p == 2, p^e, Mod[p, 4] == 1, (p^(e + 1) - 1)/(p - 1), Mod[p, 4] == 3, (p^(e + 1) + (-1)^e)/(p + 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 06 2022 *) -
PARI
a(n)=if(n<1,0,sumdiv(n,d,d*((n/d%4==1)-(n/d%4==3))))
-
PARI
{a(n)=local(A,p,e); if(n<2, n==1, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, p^e, if(p%4==1, (p^(e+1)-1)/(p-1), (p^(e+1)+(-1)^e)/(p+1)))))) } /* Michael Somos, May 02 2005 */ -
PARI
a(n)=if(n<1,0,polcoeff(sum(k=1,n,k*x^k/(1+x^(2*k)),x*O(x^n)),n))
Formula
G.f.: Sum_{n>=1} n*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
L.g.f.: Sum_{k>=1} arctan(x^k). - Ilya Gutkovskiy, Dec 16 2019
O.g.f.: Sum_{n >= 1} (-1)^(n+1) * x^(2*n-1)/(1 - x^(2*n-1))^2. - Peter Bala, Jan 04 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 1 (mod 4)} 1/(1-1/p^2) * Product_{primes p == 3 (mod 4)} 1/(1+1/p^2) = (1/2) * A175647 / A243381 = A006752/2 = 0.4579827970... . - Amiram Eldar, Nov 06 2022, Nov 05 2023
a(n) = Sum_{d|n} (n/d)*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024
Comments