cp's OEIS Frontend

A035187 Sum over divisors d of n of Kronecker symbol (5|d).

%I A035187 #50 Oct 11 2022 06:27:58
%S A035187 1,0,0,1,1,0,0,0,1,0,2,0,0,0,0,1,0,0,2,1,0,0,0,0,1,0,0,0,2,0,2,0,0,0,
%T A035187 0,1,0,0,0,0,2,0,0,2,1,0,0,0,1,0,0,0,0,0,2,0,0,0,2,0,2,0,0,1,0,0,0,0,
%U A035187 0,0,2,0,0,0,0,2,0,0,2,1,1,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,2,1,2,0,0,0,0
%N A035187 Sum over divisors d of n of Kronecker symbol (5|d).
%C A035187 Let tau be the golden ratio (1+sqrt(5))/2; let zetaQ(tau)(s)=sum(1/(Z(tau):a)^s) the Dedekind zeta function where a runs through the nonzero ideals of Z(tau) and where (Z(tau):a) is the norm of a; then zetaQ(tau)(s)=sum(n>=1,a(n)/n^s). - _Benoit Cloitre_, Dec 29 2002
%C A035187 First occurrence of k beginning at zero, or 0 if not yet known: 2, 1, 11, 121, 209, 14641, 2299, 1771561, 6061, 43681, 278179, 0, 66671, 0, 33659659, 5285401, 187891, 0, 1266749, 0, 8067191, 639533521, 0, 0, 2066801, 0, 0, 36735721, 976130111, 0, 153276629, 0, 7703531, 0, 0, 0, 39269219, 0, 0, 0, 250082921, 0, 0, 0, 0, 0, 0, 0, 84738841, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 454508329, ..., .
%C A035187 If k is prime, the 0 above can be replaced by the smallest p^(k-1) with p a prime == {1,4} (mod 5), which is p=11. This follows from the multiplicative formula. - _R. J. Mathar_, Apr 02 2011
%C A035187 The terms often equal A001157(n) mod 5; the exceptions are at n = 2299, 3509, 3751, 3971, 4961, 6061, 6479, ... - _R. J. Mathar_, Apr 02 2011
%C A035187 Coefficients of Dedekind zeta function for the quadratic number field of discriminant 5. See A002324 for formula and Maple code. - _N. J. A. Sloane_, Mar 22 2022
%H A035187 G. C. Greubel, <a href="/A035187/b035187.txt">Table of n, a(n) for n = 1..5000</a>
%H A035187 M. Baake, <a href="http://www.mathematik.uni-bielefeld.de/baake/preprints.html">Algebra, Combinatorics and Number Theory</a>.
%H A035187 M. Baake and R. V. Moody, <a href="https://arxiv.org/abs/math/9904028">Similarity submodules and root systems in four dimensions</a>, arXiv:math/9904028 [math.MG], 1999.
%H A035187 M. Baake and R. V. Moody, <a href="http://dx.doi.org/10.4153/CJM-1999-057-0">Similarity submodules and root systems in four dimensions</a>, Canad. J. Math. 51 (1999), 1258-1276.
%F A035187 Dirichlet g.f.: Product_p ( (1 - p^(-s)) (1 - Kronecker( 5, p)*p^(-s)) )^(-1).
%F A035187 Sum_{k=1..n} a(k) is asymptotic to c*n where c=2*log(tau)/sqrt(5) (A086466).
%F A035187 Multiplicative with a(5^e) = 1, a(p^e) = e+1 if p == 1, 4 (mod 5), a(p^e) = (1+(-1)^e)/2 if p == 2, 3 (mod 5). - _Michael Somos_, Jun 06 2005
%F A035187 Moebius transform is period 5 sequence A080891. - _Michael Somos_, Oct 29 2005
%F A035187 q-series for a(n): Sum_{n >= 1} -(-1)^nq^(n(n+1)/2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). - _Jeremy Lovejoy_, Jun 12 2009
%e A035187 G.f. = x + x^4 + x^5 + x^9 + 2*x^11 + x^16 + 2*x^19 + x^20 + x^25 + 2*x^29 + ...
%p A035187 A035187 := proc(n) local f,p; f := ifactors(n)[2] ; if nops(f) = 1 then p := op(1,f) ; if op(1,p) = 5 then 1; elif op(1,p) mod 5 in {1,4} then op(2,p)+1 ; else (1+(-1)^op(2,p))/2 ; end if; else mul(procname(op(1,p)^op(2,p) ),p=f) ; end if;
%p A035187 end proc: # _R. J. Mathar_, Apr 02 2011
%t A035187 f[n_] := Plus @@ (KroneckerSymbol[5, #] & /@ Divisors@ n); Array[f, 105] (* _Robert G. Wilson v_ *)
%t A035187 a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ 5, #] &]]; (* _Michael Somos_, Jun 12 2014 *)
%o A035187 (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 5, p) * X))[n])}; \\ _Michael Somos_, Jun 06 2005
%o A035187 (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k,1], e = A[k,2]; if( p==5, 1, if((p%5==1) || (p%5==4), e+1, !(e%2))))))}; \\ _Michael Somos_, Jun 06 2005
%o A035187 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 5, d) ) )}; \\ _Michael Somos_, Oct 29 2005
%Y A035187 Cf. A031363 (for indices of nonzero terms), A078428.
%Y A035187 Cf. A001622, A086466.
%Y A035187 Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
%Y A035187 Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
%K A035187 nonn,mult
%O A035187 1,11
%A A035187 _N. J. A. Sloane_