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A035187
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Sum over divisors d of n of Kronecker symbol (5|d).
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6
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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; internal format)
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OFFSET
| 1,11
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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)
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
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LINKS
| M. Baake, Algebra, Combinatorics and Number Theory
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.
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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)
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))). [From Jeremy Lovejoy (lovejoy(AT)liafa.jussieu.fr), Jun 12 2009]
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EXAMPLE
| x + x^4 + x^5 + x^9 + 2*x^11 + x^16 + 2*x^19 + x^20 + x^25 + 2*x^29 + ...
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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
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MATHEMATICA
| f[n_] := Plus @@ (KroneckerSymbol[5, #] & /@ Divisors@ n); Array[f, 105] (* RGWv *)
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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 */
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CROSSREFS
| Cf. A031363 (for denominators), A078428.
Sequence in context: A086014 A025437 A066032 * A033770 A101668 A141846
Adjacent sequences: A035184 A035185 A035186 * A035188 A035189 A035190
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KEYWORD
| nonn,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Benoit Cloitre, Dec 29, 2002
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