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.

Showing 1-7 of 7 results.

A353688 a(n) = n / A098988(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 6, 7, 4, 1, 2, 5, 4, 1, 1, 3, 1, 1, 6, 1, 2, 3
Offset: 0

Views

Author

Antti Karttunen, May 06 2022

Keywords

Links

Crossrefs

Cf. A098988.

Programs

  • PARI
    A098988(n) = if(0==n, 1, denominator(sumdiv(n,d, ((-1)^(d+1))/d)));
A353688(n) = (n / A098988(n));

A098987 Numerators in series expansion of log(Product_{m>=0} (1+q^m)).

Original entry on oeis.org

0, 1, 1, 4, 1, 6, 2, 8, 1, 13, 3, 12, 1, 14, 4, 8, 1, 18, 13, 20, 3, 32, 6, 24, 1, 31, 7, 40, 2, 30, 4, 32, 1, 16, 9, 48, 13, 38, 10, 56, 3, 42, 16, 44, 3, 26, 12, 48, 1, 57, 31, 24, 7, 54, 20, 72, 1, 80, 15, 60, 2, 62, 16, 104, 1, 84, 8, 68, 9, 32, 24, 72, 13, 74, 19, 124, 5, 96, 28, 80, 3, 121, 21, 84, 8, 108
Offset: 0

Views

Author

N. J. A. Sloane, Oct 24 2004

Keywords

Examples

			q + (1/2)*q^2 + (4/3)*q^3 + (1/4)*q^4 + (6/5)*q^5 + (2/3)*q^6 + (8/7)*q^7 + (1/8)*q^8 + (13/9)*q^9 + ...

Links

Crossrefs

Cf. A098988 (denominators).
Cf. A069519 (apparently the positions of 1's), A353687.

Programs

Formula

Numerators of a(n) = Sum_{d|n} ((-1)^(d+1))/d. - Ridouane Oudra, Apr 28 2019
Numerators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k * (1 - x^k)). - Ilya Gutkovskiy, Oct 03 2022

Extensions

Data section extended up to term a(85) by Antti Karttunen, May 06 2022

A353667 a(n) = n / gcd(n, A351546(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 18, 19, 10, 21, 11, 23, 6, 25, 13, 27, 1, 29, 15, 31, 32, 11, 17, 35, 36, 37, 19, 39, 4, 41, 21, 43, 11, 15, 23, 47, 12, 49, 50, 17, 26, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 64, 65, 33, 67, 34, 23, 35, 71, 72, 73, 37, 75, 19, 77, 39, 79, 40, 81, 41, 83, 3
Offset: 1

Views

Author

Antti Karttunen, May 04 2022

Keywords

Links

Crossrefs

Cf. A000203, A003961, A351546, A353666, A353668.
Differs from A098988 for the first time at n = 28, 30, 40, 60, 66, 84, 90, 102, 120, ...

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351546(n) = { my(f=factor(sigma(n)), u=A003961(n)); prod(k=1, #f~, f[k, 1]^((0!=(u%f[k, 1]))*f[k, 2])); };
A353667(n) = (n / gcd(n, A351546(n)));

Formula

a(n) = n / A353666(n) = n / gcd(n, A351546(n)).

A305006 Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).

Original entry on oeis.org

1, -1, 4, -5, 6, -2, 8, -13, 13, -3, 12, -5, 14, -4, 8, -29, 18, -13, 20, -3, 32, -6, 24, -13, 31, -7, 40, -10, 30, -4, 32, -61, 16, -9, 48, -65, 38, -10, 56, -39, 42, -16, 44, -15, 26, -12, 48, -29, 57, -31, 24, -35, 54, -20, 72, -13, 80, -15, 60, -2, 62, -16, 104, -125, 84
Offset: 1

Views

Author

Ilya Gutkovskiy, May 23 2018

Keywords

Examples

			1, -1/2, 4/3, -5/4, 6/5, -2/3, 8/7, -13/8, 13/9, -3/5, 12/11, -5/3, 14/13, -4/7, 8/5, -29/16, 18/17, -13/18, 20/19, ...

Crossrefs

Cf. A002129, A010054, A017665, A017666, A098987, A098988, A305007 (denominators).

Programs

  • Magma
    [Numerator(&+[(-1)^(d+1)*d/n: d in Divisors(n)]): n in [1..100]]; // Vincenzo Librandi, May 24 2018
  • Mathematica
    nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[x^k/(k (1 + x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Log[Product[(1 - x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Log[EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))], {x, 0, nmax}], x]]]
Numerator[Table[Sum[(-1)^(n/d + 1) 1/d, {d, Divisors[n]}], {n, 65}]]
Numerator[Table[DivisorSum[n, -(-1)^# # &]/n, {n, 65}]]
  • PARI
    a(n) = numerator(sumdiv(n, d, (-1)^(d+1)*d/n)); \\ Michel Marcus, May 24 2018

Formula

Numerators of coefficients in expansion of log(Sum_{k>=0} x^(k*(k+1)/2)) = log(Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1))).
Numerators of coefficients in expansion of log(theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.
a(n) = numerator of Sum_{d|n} (-1)^(n/d+1)/d.
a(n) = numerator of Sum_{d|n} (-1)^(d+1)*d/n.
a(n) = numerator of A002129(n)/n.
a(p) = p + 1 where p is an odd prime.

A305007 Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k*(1 + x^k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 18, 19, 2, 21, 11, 23, 6, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 35, 36, 37, 19, 39, 20, 41, 21, 43, 11, 15, 23, 47, 12, 49, 50, 17, 26, 53, 27, 55, 7, 57, 29, 59, 1, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35, 71, 72, 73, 37, 75
Offset: 1

Views

Author

Ilya Gutkovskiy, May 23 2018

Keywords

Examples

			1, -1/2, 4/3, -5/4, 6/5, -2/3, 8/7, -13/8, 13/9, -3/5, 12/11, -5/3, 14/13, -4/7, 8/5, -29/16, 18/17, -13/18, 20/19, ...

Crossrefs

Cf. A002129, A010054, A017665, A017666, A098987, A098988, A305006 (numerators).

Programs

  • Magma
    [Denominator(&+[(-1)^(d+1)*d/n: d in Divisors(n)]): n in [1..100]]; // Vincenzo Librandi, May 24 2018
  • Mathematica
    nmax = 75; Rest[Denominator[CoefficientList[Series[Sum[x^k/(k (1 + x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Log[Product[(1 - x^(2 k))/(1 - x^(2 k - 1)), {k, 1, nmax}]], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Log[EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))], {x, 0, nmax}], x]]]
Denominator[Table[Sum[(-1)^(n/d + 1) 1/d, {d, Divisors[n]}], {n, 75}]]
Denominator[Table[DivisorSum[n, -(-1)^# # &]/n, {n, 75}]]
  • PARI
    a(n) = denominator(sumdiv(n, d, (-1)^(d+1)*d/n)); \\ Michel Marcus, May 24 2018

Formula

Denominators of coefficients in expansion of log(Sum_{k>=0} x^(k*(k+1)/2)) = log(Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1))).
Denominators of coefficients in expansion of log(theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.
a(n) = denominator of Sum_{d|n} (-1)^(n/d+1)/d.
a(n) = denominator of Sum_{d|n} (-1)^(d+1)*d/n.
a(n) = denominator of A002129(n)/n.
a(p^k) = p^k where p is a prime.

A357556 a(n) is the denominator of Sum_{d|n} (-1)^(d+1) / d^2.

Original entry on oeis.org

1, 4, 9, 16, 25, 6, 49, 64, 81, 50, 121, 72, 169, 98, 45, 256, 289, 108, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 15, 961, 1024, 1089, 578, 49, 1296, 1369, 722, 1521, 800, 1681, 147, 1849, 88, 2025, 1058, 2209, 128, 2401, 2500, 2601, 1352, 2809, 243, 3025
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 03 2022

Keywords

Examples

			1, 3/4, 10/9, 11/16, 26/25, 5/6, 50/49, 43/64, 91/81, 39/50, 122/121, ...

Crossrefs

Cf. A017668, A064027, A098988, A321543, A334580, A357555 (numerators).

Programs

  • Mathematica
    Table[Sum[(-1)^(d + 1)/d^2, {d, Divisors[n]}], {n, 1, 55}] // Denominator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
  • PARI
    a(n) = denominator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022
  • Python
    from sympy import divisors
from fractions import Fraction
def a(n): return sum(Fraction((-1)**(d+1), d*d) for d in divisors(n, generator=True)).denominator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022

Formula

Denominators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k^2 * (1 - x^k)).

A380271 Denominators of coefficients in expansion of exp(-1 + 1 / Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 1008, 40320, 72576, 3628800, 39916800, 95800320, 6227020800, 3487131648, 1307674368000, 20922789888000, 2845499424768, 6402373705728000, 24329020081766400, 187146308321280000, 51090942171709440000, 224800145555521536000, 25852016738884976640000
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 18 2025

Keywords

Examples

			1, 1, 5/2, 31/6, 265/24, 2621/120, 31621/720, 85319/1008, 6574961/40320, ...

Crossrefs

Cf. A000041, A017666, A058892, A066186, A067653, A098988, A380171 (numerators).

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[-1 + 1/Product[1 - x^k, {k, 1, nmax}]], {x, 0, nmax}], x] // Denominator
b[0] = 1; b[n_] := b[n] = (1/n) Sum[k PartitionsP[k] b[n - k], {k, 1, n}]; a[n_] := Denominator[b[n]]; Table[a[n], {n, 0, 23}]

Formula

b(0) = 1, b(n) = (1/n) * Sum_{k=1..n} k * A000041(k) * b(n-k), a(n) = denominator of b(n).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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