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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.

Showing 1-3 of 3 results.

A258655 a(n) = A256357(n^2), where exp( Sum_{n>=1} A256357(n)*x^n/n ) = 1 + Sum_{n>=1} x^(n^2) + x^(2*n^2).

Original entry on oeis.org

1, 5, 7, -19, 21, 59, 57, -115, 61, 145, 111, -253, 157, 285, 147, -499, 307, 545, 343, -599, 399, 643, 553, -1501, 521, 889, 547, -1083, 813, 1759, 993, -2035, 777, 1535, 1197, -2359, 1333, 1867, 1099, -3575, 1723, 3363, 1807, -2549, 1281, 2765, 2257, -6493, 2801, 3645, 2149, -3503, 2757
Offset: 1

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Author

Paul D. Hanna, Jun 06 2015

Keywords

Comments

a(4*n) < 0 for n>=1, and a(n) is positive if n is not divisible by 4 (conjecture).

Examples

			L.g.f.: L(x) = x + 5*x^2/2 + 7*x^3/3 - 19*x^4/4 + 21*x^5/5 + 59*x^6/6 + 57*x^7/7 - 115*x^8/8 + 61*x^9/9 + 145*x^10/10 +...+ A256357(n^2)*x^n/n +...
where
exp(L(x)) = 1 + x + 3*x^2 + 5*x^3 + 2*x^4 + 10*x^5 + 13*x^6 + 23*x^7 + 43*x^8 + 57*x^9 + 66*x^10 +...+ A258656(n)*x^n +...

Links

Crossrefs

Cf. A258656, A256357.

Programs

  • PARI
    {a(n) = local(L=x); L = log(1 + sum(k=1, n+1, x^(k^2) + x^(2*k^2)) +x*O(x^(n^2))); n^2*polcoeff(L, n^2)}
for(n=1, 70, print1(a(n), ", "))
  • PARI
    {a(n) = -sigma(n^2) + sumdiv(n^2,d, if(d%4==2,d)) + 2*sumdiv(n^2,d, if((d%8)%3==1,d))}
for(n=1,70, print1(a(n),", "))

Formula

a(n) = -sigma(n^2) + [Sum_{d|n^2, d==2 (mod 4)} d] + [Sum_{d|n^2, d==1,4,7 (mod 8)} 2*d].

A258656 O.g.f.: exp( Sum_{n>=1} A256357(n^2)*x^n/n ), where exp( Sum_{n>=1} A256357(n)*x^n/n ) = 1 + Sum_{n>=1} x^(n^2) + x^(2*n^2).

Original entry on oeis.org

1, 1, 3, 5, 2, 10, 13, 23, 43, 57, 66, 96, 183, 229, 375, 509, 619, 883, 1395, 1947, 2487, 3603, 4627, 6273, 8934, 12432, 15637, 20943, 29147, 37613, 50296, 67870, 88542, 113240, 153682, 201900, 257125, 338397, 446354, 570098, 734576, 966634, 1234879, 1574763, 2048746, 2634002, 3322639, 4268611
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2015

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 5*x^3 + 2*x^4 + 10*x^5 + 13*x^6 + 23*x^7 +...
where
log(A(x)) = x + 5*x^2/2 + 7*x^3/3 - 19*x^4/4 + 21*x^5/5 + 59*x^6/6 + 57*x^7/7 - 115*x^8/8 + 61*x^9/9 + 145*x^10/10 +...+ A256357(n^2)*x^n/n +...

Links

Crossrefs

Cf. A258655, A256357.

Programs

  • PARI
    {A258655(n) = local(L=x); L = log(1 + sum(k=1, n+1, x^(k^2) + x^(2*k^2)) +x*O(x^(n^2))); n^2*polcoeff(L, n^2)}
{a(n) = polcoeff( exp( sum(k=1,n+1, A258655(k)*x^k/k) +x*O(x^n) ), n)}
for(n=1, 121, print1(a(n), ", "))
  • PARI
    /* Much faster: */
{A258655(n) = -sigma(n^2) + sumdiv(n^2, d, if(d%4==2, d)) + 2*sumdiv(n^2, d, if((d%8)%3==1, d))}
{a(n) = polcoeff( exp( sum(k=1,n+1, A258655(k)*x^k/k) +x*O(x^n) ), n)}
for(n=1,121, print1(a(n), ", "))

A258328 L.g.f.: log(1 + Sum_{n>=1} x^(n^2) + x^(3*n^2) ).

Original entry on oeis.org

1, -1, 4, -1, 1, -4, 1, -1, 13, -11, 12, -16, 14, -15, 19, -1, 1, -13, 1, -11, 25, -12, 24, -40, 26, -14, 40, -15, 1, -29, 1, -1, 48, -35, 36, -61, 38, -39, 56, -11, 1, -39, 1, -12, 73, -24, 48, -88, 50, -36, 55, -14, 1, -40, 12, -15, 61, -59, 60, -101, 62, -63, 97, -1, 14, -48, 1, -35, 96, -60, 72, -157, 74, -38, 119, -39, 12, -56, 1, -11, 121, -83, 84, -135, 86, -87, 91, -12, 1, -83, 14, -24, 97, -48, 96, -184, 98, -64, 156, -36, 1, -89, 1, -14, 180, -107, 108, -196, 110, -132, 152, -15, 1, -99, 24, -59, 182, -60, 120, -245, 133
Offset: 1

Views

Author

Paul D. Hanna, Jun 03 2015

Keywords

Examples

			L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 - 4*x^6/6 + x^7/7 - x^8/8 + 13*x^9/9 - 11*x^10/10 + 12*x^11/11 - 16*x^12/12 + 14*x^13/13 - 15*x^14/14 + 19*x^15/15 - x^16/16 +...+ a(n)*x^n/n +...
where
exp(L(x)) = 1 + x + x^3 + x^4 + x^9 + x^12 + x^16 + x^25 + x^27 + x^36 + x^48 + x^49 + x^64 + x^75 + x^81 + x^100 + x^108 +...+ x^(n^2) + x^(3*n^2) +...
Note that for n>1, a(n) = +1 at positions:
[5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 113, 127, ...];
which appears to be A003630 (primes p such that 3 is not a square mod p).

Links

Crossrefs

Cf. A256357, A003630, A038875.

Programs

  • PARI
    {a(n) = local(L=x); L = log(1 + sum(k=1,sqrtint(n+1), x^(k^2) + x^(3*k^2)) +x*O(x^n)); n*polcoeff(L,n)}
for(n=1,121, print1(a(n),", "))

Formula

a(n) = -1 iff n = 2^k for k>=1 [conjecture].
a(p) = +1 for primes p such that 3 is not a square mod p (A003630), and a(n) = +1 nowhere else except at n=0 [conjecture].
Showing 1-3 of 3 results.

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