A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.
1, 2, 3, 2, 1, 2, 2, 4, 2, 2, 1, 0, 4, 2, 3, 2, 2, 4, 0, 2, 2, 0, 4, 2, 3, 0, 2, 6, 2, 2, 1, 2, 0, 2, 2, 2, 2, 4, 2, 0, 4, 4, 4, 0, 1, 2, 0, 4, 2, 0, 2, 2, 5, 2, 0, 2, 2, 4, 4, 2, 0, 2, 4, 2, 2, 0, 4, 0, 0, 2, 3, 2, 4, 2, 0, 4, 0, 6, 2, 4, 1, 0, 4, 2, 2, 2, 2, 0, 0, 2, 0, 2, 8, 2, 2, 0, 2, 4, 0, 4, 2, 2, 3, 2, 2
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ... G.f. = q + 2*q^13 + 3*q^25 + 2*q^37 + q^49 + 2*q^61 + 2*q^73 + 4*q^85 + 2*q^97 + ...
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- John Cerkan, Table of n, a(n) for n = 0..10000
- J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Programs
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Maple
series(mul( ( (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2, n = 1..100), q, 101): seq(coeftayl(%, q = 0, n), n = 0..100); # Peter Bala, Jan 05 2025
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Mathematica
ed[n_]:=Module[{divs=Divisors[12n+1]},Count[divs,?(Mod[#,4] == 1&)]- Count[divs,?(Mod[#,4]==3&)]]; Array[ed,110,0] (* Harvey P. Dale, Jul 01 2012 *) a[ n_] := If[ n < 0, 0, With[ {m = 12 n + 1}, Sum[ KroneckerSymbol[ 4, d], {d, Divisors[m]}]]]; (* Michael Somos, Apr 23 2014 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^6]))^2, {x, 0, n}]; (* Michael Somos, Apr 23 2014 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2])^2, {x, 0, n}]; (* Michael Somos, May 25 2015 *) -
PARI
{a(n) = if( n<0, 0, n = 12*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 19 2005 */ -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Jun 02 2012 */
Formula
Expansion of (phi(-x^3) / chi(-x))^2 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/12) * (eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)))^2 in powers of q. - Michael Somos, Sep 19 2005
Euler transform of period 6 sequence [ 2, 0, -2, 0, 2, -2, ...]. - Michael Somos, Sep 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258279. - Michael Somos, May 25 2015
From Michael Somos, Jun 02 2012: (Start)
a(n) = A(12*n+1) where A = A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113652, A121450, A122864, A125061, A129448, A132004, A134013, A134015, A138746, A138950, A138952, A163746. (End)
From Michael Somos, May 25 2015: (Start)
G.f.: (Sum_{k = -oo..oo} x^(k*(3*k-1)/2))^2. - Ilya Gutkovskiy, Aug 14 2017
G.f.: ( Product_{n >= 1} (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2. - Peter Bala, Jan 05 2025
Comments