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.

A339354 G.f.: Sum_{k>=1} k^3 * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, 9, 1, 9, 1, 9, 1, 36, 1, 9, 28, 9, 1, 36, 1, 73, 28, 9, 1, 100, 1, 9, 28, 73, 1, 161, 1, 73, 28, 9, 126, 100, 1, 9, 28, 198, 1, 252, 1, 73, 153, 9, 1, 316, 1, 134, 28, 73, 1, 252, 126, 416, 28, 9, 1, 441, 1, 9, 371, 73, 126, 252, 1, 73, 28, 477, 1, 828, 1, 9, 153, 73, 344
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Comments

Sum of cubes of divisors of n that are smaller than sqrt(n).

Links

Crossrefs

Cf. A001158, A056924, A070039, A276634, A280375, A339353.

Programs

  • Mathematica
    nmax = 77; CoefficientList[Series[Sum[k^3 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^3 &, # < Sqrt[n] &], {n, 77}]
  • PARI
    a(n) = sumdiv(n, d, if (d^2 < n, d^3)); \\ Michel Marcus, Dec 02 2020

A347142 Sum of 4th powers of divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 17, 1, 17, 1, 17, 1, 98, 1, 17, 82, 17, 1, 98, 1, 273, 82, 17, 1, 354, 1, 17, 82, 273, 1, 723, 1, 273, 82, 17, 626, 354, 1, 17, 82, 898, 1, 1394, 1, 273, 707, 17, 1, 1650, 1, 642, 82, 273, 1, 1394, 626, 2674, 82, 17, 1, 2275, 1, 17, 2483, 273, 626
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 19 2021

Keywords

Crossrefs

Cf. A001159, A056924, A070039, A279363, A339353, A339354, A347143.

Programs

  • Mathematica
    Table[DivisorSum[n, #^4 &, # < Sqrt[n] &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[k^4 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A347142(n) = { my(s=0); fordiv(n,d,if((d^2)>=n,return(s)); s += (d^4)); }; \\ Antti Karttunen, Aug 19 2021

Formula

G.f.: Sum_{k>=1} k^4 * x^(k*(k + 1)) / (1 - x^k).

A347161 Sum of squares of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 35, 1, 1, 10, 1, 26, 10, 1, 1, 10, 26, 1, 10, 1, 1, 35, 1, 1, 10, 1, 26, 10, 1, 1, 10, 26, 50, 10, 1, 1, 35, 1, 1, 59, 1, 26, 10, 1, 1, 10, 75, 1, 10, 1, 1, 35, 1, 50, 10, 1, 26
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 20 2021

Keywords

Crossrefs

Cf. A001157, A050999, A333805, A333807, A339353, A347162.

Programs

  • Mathematica
    Table[DivisorSum[n, #^2 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(2 k - 1)^2 x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = my(r=sqrt(n)); sumdiv(n, d, if ((d%2) && (dMichel Marcus, Aug 21 2021

Formula

G.f.: Sum_{k>=1} (2*k - 1)^2 * x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 4, 0, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 53, 9, 4, 0, 38, 0, 4, 58, 4, 25, 13, 0, 4, 9, 78, 0, 13, 0, 4, 34, 4, 49, 13, 0, 29
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 20 2021

Keywords

Crossrefs

Cf. A001157, A005063, A005066, A098002, A333806, A333808, A339353, A347157, A347158.

Programs

  • Mathematica
    Table[DivisorSum[n, #^2 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[Prime[k]^2 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

G.f.: Sum_{k>=1} prime(k)^2 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

A373032 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k*(k+1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, -3, 1, -3, 1, -3, 1, 6, 1, -3, 10, -3, 1, 6, 1, -19, 10, -3, 1, -10, 1, -3, 10, -19, 1, 31, 1, -19, 10, -3, 26, -10, 1, -3, 10, 6, 1, -30, 1, -19, 35, -3, 1, -46, 1, 22, 10, -19, 1, -30, 26, 30, 10, -3, 1, -21, 1, -3, 59, -19, 26, -30, 1, -19, 10, 71
Offset: 1

Views

Author

Ilya Gutkovskiy, May 20 2024

Keywords

Crossrefs

Cf. A056924, A064027, A070039, A321543, A333809, A333810, A339353, A344300, A372625, A373031.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(d+1) * d^2.
Showing 1-5 of 5 results.

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