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 1-5 of 5 results.

A175926 Sum of divisors of cubes.

Original entry on oeis.org

1, 15, 40, 127, 156, 600, 400, 1023, 1093, 2340, 1464, 5080, 2380, 6000, 6240, 8191, 5220, 16395, 7240, 19812, 16000, 21960, 12720, 40920, 19531, 35700, 29524, 50800, 25260, 93600, 30784, 65535, 58560, 78300, 62400, 138811, 52060, 108600, 95200
Offset: 1

Views

Author

Zak Seidov, Oct 19 2010

Keywords

Comments

The Mobius transform of the sequence is 1, 14, 39 ,112, 155,..., which equals the sequence defined by n*A160889(n). - R. J. Mathar, Apr 15 2011
Zhi-Wei Sun noted that the first 10^7 terms are pairwise distinct, but Noam D. Elkies found that a(48142241) = a(48374911), a(384422506) = a(403764207) and so on. - Zhi-Wei Sun, Jan 08 2014

Links

Crossrefs

Cf. sigma(n^k): A000203 (k=1), A065764 (k=2), this sequence (k=3), A202994 (k=4), A203556 (k=5).
Cf. A000578, A013662, A160889.

Programs

  • Magma
    [ SumOfDivisors(n^3) : n in [1..100]]; // Vincenzo Librandi, Apr 14 2011
  • Mathematica
    DivisorSigma[1,#]&/@((Range[40])^3) (* Harvey P. Dale, Aug 30 2015 *)
f[p_, e_] := (p^(3*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)
  • PARI
    a(n) = sigma(n^3); \\ Amiram Eldar, Nov 05 2022
  • Python
    from math import prod
from sympy import factorint
def A175926(n): return prod((p**(3*e+1)-1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Oct 25 2023

Formula

a(n) = A000203(n^3). - R. J. Mathar, Mar 31 2011
Multiplicative with a(p^e) = (p^(3e+1)-1)/(p-1). - R. J. Mathar, Mar 31 2011
Sum_{k>=1} 1/a(k) = 1.11535899887110289127674868460900333554265894187008102863022551119560512446... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = 0.4732277044... . - Amiram Eldar, Nov 05 2022

A202993 G.f.: A(x) = exp( Sum_{n>=1} sigma(n^4)*x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 1, 16, 56, 296, 1052, 4952, 17292, 70512, 249712, 931226, 3212690, 11399590, 38331770, 130310820, 428389292, 1408697596, 4524980036, 14486512316, 45558807176, 142488702483, 439559056419, 1347096766984, 4082169772704, 12286806024269, 36629267989081
Offset: 0

Views

Author

Paul D. Hanna, Dec 27 2011

Keywords

Comments

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 16*x^2 + 56*x^3 + 296*x^4 + 1052*x^5 + 4952*x^6 +...
log(A(x)) = x + 31*x^2/2 + 121*x^3/3 + 511*x^4/4 + 781*x^5/5 + 3751*x^6/6 + 2801*x^7/7 + 8191*x^8/8 +...+ A202994(n)*x^n/n +...

Links

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A202994, A156304.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sigma(m^4)*x^m/m)+x*O(x^n)), n)}
  • PARI
    {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k^4)*a(n-k)))}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k^4) * a(n-k) for n>0, with a(0)=1.
log(a(n)) ~ 5 * 3^(1/5) * c^(1/5) * n^(4/5) / 2^(7/5), where c = Product_{primes p} (p*(1 + p + p^2 + p^4) / (p^5 - 1)) = 1.9202928959802946010362130828... - Vaclav Kotesovec, Nov 01 2024

A203556 a(n) = sigma(n^5).

Original entry on oeis.org

1, 63, 364, 2047, 3906, 22932, 19608, 65535, 88573, 246078, 177156, 745108, 402234, 1235304, 1421784, 2097151, 1508598, 5580099, 2613660, 7995582, 7137312, 11160828, 6728904, 23854740, 12207031, 25340742, 21523360, 40137576, 21243690, 89572392, 29583456, 67108863
Offset: 1

Views

Author

Paul D. Hanna, Jan 03 2012

Keywords

Comments

a(n) modulo 6 begins: [1,3,4,1,0,0,0,3,1,0,0,4,0,0,0,1,0,3,0,0,0,0,0,0,1,0,...], in which positions of nonzero residues seem related to squares.

Examples

			L.g.f.: L(x) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...
where the g.f. of A203557 begins:
exp(L(x)) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...

Links

Crossrefs

Cf. A203557 (exp), A000203 (sigma), A000584, A013664.
Variants: A065764, A175926, A202994.

Programs

  • Mathematica
    f[p_, e_] := (p^(5*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 09 2020 *)
DivisorSigma[1,Range[40]^5] (* Harvey P. Dale, Dec 05 2021 *)
  • PARI
    a(n) = sigma(n^5)

Formula

Logarithmic derivative of A203557.
Multiplicative with a(p^e) = (p^(5*e+1)-1)/(p-1) for prime p. - Andrew Howroyd, Jul 23 2018
From Amiram Eldar, Nov 05 2022: (Start)
a(n) = A000203(A000584(n)) = A000203(n^5).
Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3220880186... . (End)

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A344327 Number of divisors of n^4.

Original entry on oeis.org

1, 5, 5, 9, 5, 25, 5, 13, 9, 25, 5, 45, 5, 25, 25, 17, 5, 45, 5, 45, 25, 25, 5, 65, 9, 25, 13, 45, 5, 125, 5, 21, 25, 25, 25, 81, 5, 25, 25, 65, 5, 125, 5, 45, 45, 25, 5, 85, 9, 45, 25, 45, 5, 65, 25, 65, 25, 25, 5, 225, 5, 25, 45, 25, 25, 125, 5, 45, 25, 125, 5, 117, 5, 25, 45, 45, 25, 125, 5, 85, 17, 25
Offset: 1

Views

Author

Seiichi Manyama, May 15 2021

Keywords

Links

Crossrefs

Column k=4 of A343656.
Cf. A000005, A000583, A202994.

Programs

  • Mathematica
    Table[DivisorSigma[0, n^4], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)
  • PARI
    a(n) = numdiv(n^4);
  • PARI
    a(n) = prod(k=1, #f=factor(n)[, 2], 4*f[k]+1);
  • PARI
    a(n) = sumdiv(n, d, 4^omega(d));
  • PARI
    my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 4^omega(k)*x^k/(1-x^k)))
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021
  • PARI
    for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

Formula

a(n) = A000005(A000583(n)).
Multiplicative with a(p^e) = 4*e+1.
a(n) = Sum_{d|n} 4^omega(d).
G.f.: Sum_{k>=1} 4^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s). - Vaclav Kotesovec, May 15 2021
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)). - Vaclav Kotesovec, Aug 20 2021

A320425 Numbers k such that sigma(sigma(k^4)) == 0 (mod k^2).

Original entry on oeis.org

1, 2, 4, 8, 16, 19, 38, 76, 152, 304, 608, 1216, 1824, 3744, 3840, 4864, 6400, 7904, 11520, 14592, 19200, 21888, 23712, 24320, 25536, 32768, 33696, 34560, 43776, 71136, 72960, 80640, 102144, 103680, 114688, 121600, 134400, 134784, 213408, 218880, 306432, 311296, 364800, 403200
Offset: 1

Views

Author

Robert G. Wilson v, Jan 08 2019

Keywords

Comments

Inspired by Allan C. Wechsler in seqfan list, Jan 07 2019.
Are 1 and 19 the only odd terms?

Links

Crossrefs

Cf. A202994 (sigma(n^4)).

Programs

  • Mathematica
    fQ[n_] := Mod[DivisorSigma[1, DivisorSigma[1, n^4]], n^2] == 0; Select[Range@ 476671, fQ]
  • PARI
    isok(n) = (sigma(sigma(n^4)) % n^2) == 0; \\ Michel Marcus, Jan 09 2019
Showing 1-5 of 5 results.

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