OFFSET
1,11
COMMENTS
Let phi be the golden ratio (1+sqrt(5))/2; let zetaQ(phi)(s) = Sum(1/(Z(phi):a)^s) the Dedekind zeta function where a runs through the nonzero ideals of Z(phi) and where (Z(phi):a) is the norm of a; then zetaQ(phi)(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
G. C. Greubel, Table of n, a(n) for n = 1..5000
M. Baake, Algebra, Combinatorics and Number Theory.
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(phi)/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
Moebius transform gives A080891.
Cf. A038872 (primes not inert in Q(sqrt(5))), A045468 (primes decomposing), A003631 (primes remaining inert), A042993 (primes not decomposing).
Dedekind zeta functions for imaginary quadratic number fields of discriminants D = -3..-47, -67, -163: A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, A035167, A192013, A035159, A035155, A035151, A035180, A035147, A035143, A318982, A318983.
KEYWORD
nonn,easy,mult
AUTHOR
STATUS
approved