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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 Michael 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 for e >= 0,    a(p^e) = 0 if p == 3 (mod 4) for e > 0,    a(p^e) = 2 if p == 1 (mod 4) for e > 0. (corrected by Werner Schulte, Dec 12 2020. a(2*n) = a(n). a(3*n) = a(4*n + 3) = 0. 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-François 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 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 September 24 22:46 EDT 2021. Contains 347651 sequences. (Running on oeis4.)