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A035187 Sum over divisors d of n of Kronecker symbol (5|d). 35
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,11
COMMENTS
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
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, ..., .
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
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
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
LINKS
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.
FORMULA
Dirichlet g.f.: Product_p ( (1 - p^(-s)) (1 - Kronecker( 5, p)*p^(-s)) )^(-1).
Sum_{k=1..n} a(k) is asymptotic to c*n where c=2*log(tau)/sqrt(5) (A086466).
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
Moebius transform is period 5 sequence A080891. - Michael Somos, Oct 29 2005
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
EXAMPLE
G.f. = x + x^4 + x^5 + x^9 + 2*x^11 + x^16 + 2*x^19 + x^20 + x^25 + 2*x^29 + ...
MAPLE
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;
end proc: # R. J. Mathar, Apr 02 2011
MATHEMATICA
f[n_] := Plus @@ (KroneckerSymbol[5, #] & /@ Divisors@ n); Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ 5, #] &]]; (* Michael Somos, Jun 12 2014 *)
PROG
(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
(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
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 5, d) ) )}; \\ Michael Somos, Oct 29 2005
CROSSREFS
Cf. A031363 (for indices of nonzero terms), A078428.
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.
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.
Sequence in context: A086014 A025437 A066032 * A291147 A278929 A363738
KEYWORD
nonn,mult
AUTHOR
STATUS
approved

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Last modified March 28 07:30 EDT 2024. Contains 371235 sequences. (Running on oeis4.)