The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A226333 Expansion of (E_4(q) - E_4(q^5)) / 240 in powers of q where E_4 is an Eisenstein series. 1
 1, 9, 28, 73, 125, 252, 344, 585, 757, 1125, 1332, 2044, 2198, 3096, 3500, 4681, 4914, 6813, 6860, 9125, 9632, 11988, 12168, 16380, 15625, 19782, 20440, 25112, 24390, 31500, 29792, 37449, 37296, 44226, 43000, 55261, 50654, 61740, 61544, 73125, 68922, 86688 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 1..10000 Michael Somos, Introduction to Ramanujan theta functions. Eric Weisstein's World of Mathematics, Ramanujan Theta Functions. FORMULA Expansion of q * (f(-q) * f(-q^5))^4 + 13 * q^2 * (f(-q^5)^5 / f(-q))^2 in powers of q where f() is a Ramanujan theta function. a(n) is multiplicative with a(p^e) = p^(3*e) if p=5, else a(p^e) = (p^(3*(e+1)) - 1) / (p^3 - 1). G.f.: Sum_{k>0} k^3 * x^k * (1 - x^(4*k)) / ((1 - x^k) * (1 - x^(5*k))). a(n) = A004009(n) if n is not divisible by 5, else a(n) = 5^3 * a(n/5). From Amiram Eldar, Sep 12 2023: (Start) Dirichlet g.f.: (1 - 1/5^s) * zeta(s-3) * zeta(s). Sum_{k=1..n} a(k) ~ c * n^4, where c = 26*Pi^4/9375 = 0.270147... . (End) EXAMPLE q + 9*q^2 + 28*q^3 + 73*q^4 + 125*q^5 + 252*q^6 + 344*q^7 + 585*q^8 + 757*q^9 + ... MATHEMATICA a[ n_] := If[ n < 1, 0, DivisorSigma[ 3, n] - If[ Divisible[ n, 5], DivisorSigma[ 3, n/5], 0]] a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[q^5])^4 + 13 q^2 ( QPochhammer[q^5]^5 / QPochhammer[ q])^2, {q, 0, n}] PROG (PARI) {a(n) = if( n<1, 0, sigma( n, 3) - if( n%5, 0, sigma( n/5, 3)))} (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^5 + A))^4 + 13 * x * (eta(x^5 + A)^5 / eta(x + A))^2, n))} (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, p^(3*e), (p^(3*e+3) - 1) / (p^3 - 1)))))} CROSSREFS Cf. A000122, A000700, A004009, A010054, A121373. Sequence in context: A009255 A062451 A065959 * A017669 A277065 A001158 Adjacent sequences: A226330 A226331 A226332 * A226334 A226335 A226336 KEYWORD nonn,easy,mult AUTHOR Michael Somos, Jun 04 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 14 05:17 EDT 2024. Contains 373393 sequences. (Running on oeis4.)