cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A378009 a(n) = b(7*n+1), with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo 7; 7th column of A378007.

Original entry on oeis.org

1, 2, 0, 0, 6, 0, 6, 0, 0, 3, 6, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 3, 0, 0, 0, 6, 0, 6, 0, 0, 12, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 6, 12, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, 6, 0, 6
Offset: 0

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Author

Jianing Song, Nov 14 2024

Keywords

Examples

			Write w = exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2, then (1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + ...)*(1 + w/2^s + (w+1)/3^s - (w+1)/4^s - w/5^s - 1/6^s + ...)*(1 - (w+1)/2^s + w/3^s + w/4^s - (w+1)/5^s + 1/6^s + ...)*(1 + 1/2^s - 1/3^s + 1/4^s - 1/5^s - 1/6^s + ...)*(1 + w/2^s - (w+1)/3^s - (w+1)/4^s + w/5^s + 1/6^s + ...)*(1 - (w+1)/2^s - w/3^s + w/4^s + (w+1)/5^s - 1/6^s + ...) = 1 + 2/8^s + 6/29^s + 6/43^s + 3/64^s + 6/71^s + ...
		

Crossrefs

Cf. A378007.

Programs

  • PARI
    A378009(n) = {
    my(f = factor(7*n+1), res = 1); for(i=1, #f~,
    if(f[i,1] % 7 == 1, res *= binomial(f[i,2]+5, 5));
    if(f[i,1] % 7 == 6, if(f[i,2] % 2 == 0, res *= binomial(f[i,2]/2+2, 2), return(0)));
    if(f[i,1] % 7 == 2 || f[i,1] % 7 == 4, if(f[i,2] % 3 == 0, res *= f[i,2]/3+1, return(0)));
    if(f[i,1] % 7 == 3 || f[i,1] % 7 == 5, if(f[i,2] % 6 != 0, return(0))));
    res; }

Formula

a(n) = b(7*n+1), where {b(n)} is multiplicative with:
- b(7^e) = 0;
- for p == 1 (mod 7), b(p^e) = binomial(e+5,5) = (e+5)*(e+4)*(e+3)*(e+2)*(e+1)/120;
- for p == 6 (mod 7), b(p^e) = binomial(e/2+2,2) = (e/2+2)*(e/2+1)/2 if e is even, and 0 otherwise;
- for p == 2, 4 (mod 7), b(p^e) = e/3 + 1 if 3 divides e, and 0 otherwise;
- for p == 3, 5 (mod 7), b(p^e) = 1 if 6 divides e, and 0 otherwise.