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A091400 a(n) = Product_{ odd primes p | n } (1 + Legendre(-1,p) ). 5
1, 1, 0, 1, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

REFERENCES

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2) (but without the restriction that a(4k) = 0).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Here we use the definition that Legendre(-1, 2) = 0, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4. This is Shimura's definition, which is different from Maple's.

Moebius transform is period 36 sequence [1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...]. - Michael Somos, Apr 19 2007

Expansion of (phi(q)^2 - phi(q^9)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

a(n) is multiplicative with a(2^e) = 1, a(3^e) = 0^e, a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).

a(2*n) = a(n). a(3*n) = a(4*n + 3) = 0.

a(n) = abs(A129448(n)). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).

a(n) = Sum_{d|n} b(d)*(-1)^bigomega(d)*moebius(d) where b(2n)=0 and b(2n+1)=(-1)^n. - Benoit Cloitre, Apr 17 2016

G.f.: ((Sum_{k in Z} x^k^2)^2 - (Sum_{k in Z} x^(9*k^2))^2) / 4. - Michael Somos, Jan 26 2017

EXAMPLE

G.f. = x + x^2 + x^4 + 2*x^5 + x^8 + 2*x^10 + 2*x^13 + x^16 + 2*x^17 + 2*x^20 + ...

MAPLE

with(numtheory): A091400 := proc(n) local i, t1, t2; t1 := ifactors(n)[2]; t2 := 1; for i from 1 to nops(t1) do if t1[i][1] > 2 then t2 := t2*(1+legendre(-1, t1[i][1])); fi; od: t2; end;

with(numtheory): seq(mul(1+legendre(-1, p), p in select(isprime, divisors(n) minus {2})), n=1..105); # Peter Luschny, Apr 20 2016

MATHEMATICA

Legendre[-1, p_] := Which[p==2, 0, Mod[p, 4]==1, 1, True, -1]; a[1] = 1; a[n_] := Times @@ (Legendre[-1, #] + 1&) /@ FactorInteger[n][[All, 1]]; Array[a, 105] (* Jean-Fran├žois Alcover, Dec 01 2015 *)

Join[{1}, Table[Product[1+JacobiSymbol[-1, p], {p, Complement[FactorInteger[n][[All, 1]], {2}]}], {n, 2, 105}]] (* Peter Luschny, Apr 20 2016 *)

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 - EllipticTheta[ 3, 0, q^9]^2) / 4, {q, 0, n}]; (* Michael Somos, Jan 26 2017 *)

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-9, d) * kronecker(36, n/d)))}; /* Michael Somos, Jan 26 2017 */

(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, (-1)^bigomega(d)*moebius(d)*if(d%2, (-1)^(d\2), 0))} /* Benoit Cloitre, Apr 17 2016 */

CROSSREFS

Cf. A091379, A122856, A122865, A129448.

Sequence in context: A329343 A063725 A084888 * A129448 A239003 A123759

Adjacent sequences:  A091397 A091398 A091399 * A091401 A091402 A091403

KEYWORD

nonn,mult,changed

AUTHOR

N. J. A. Sloane, Mar 02 2004

EXTENSIONS

Definition clarified by Peter Luschny, Apr 20 2016

STATUS

approved

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)