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.

A279363 Sum of 4th powers of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 17, 1, 98, 1, 273, 82, 642, 1, 1650, 1, 2418, 707, 4369, 1, 7955, 1, 10898, 2483, 14658, 1, 26482, 626, 28578, 6643, 41090, 1, 62644, 1, 69905, 14723, 83538, 3027, 133923, 1, 130338, 28643, 174994, 1, 236692, 1, 249170, 57893, 279858, 1, 423794, 2402, 401267, 83603, 485810, 1, 644372, 15267, 659842, 130403, 707298, 1, 1053636
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 10 2016

Keywords

Examples

			a(10) = 1^4 + 2^4 + 5^4 = 642, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^4 = 1, because 11 has 1 proper divisor {1}.

Links

Crossrefs

Cf. A000583, A001159.
Cf. A001065, A067558, A276634, A279364.

Programs

  • Mathematica
    Table[DivisorSigma[4, n] - n^4, {n, 60}]
  • PARI
    for(n=1, 60, print1(sigma(n, 4) - n^4,", ")) \\ Indranil Ghosh, Mar 18 2017
  • Python
    from sympy.ntheory import divisor_sigma
print([divisor_sigma(n,4) - n**4 for n in range(1,61)]) # Indranil Ghosh, Mar 18 2017

Formula

a(n) = 1 if n is prime.
a(p^k) = (p^(4*k) - 1)/(p^4 - 1) when p is prime.
Dirichlet g.f.: zeta(s-4)*(zeta(s) - 1).
a(n) = A001159(n) - A000583(n).
G.f.: -x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5 + Sum_{k>=1} k^4 x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 18 2017
Sum_{k=1..n} a(k) ~ (Zeta(5) - 1)*n^5 / 5. - Vaclav Kotesovec, Feb 02 2019

A321258 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 0, 1, 1, 5, 1, 3, 0, 1, 1, 9, 1, 6, 1, 0, 1, 1, 17, 1, 14, 1, 3, 0, 1, 1, 33, 1, 36, 1, 7, 2, 0, 1, 1, 65, 1, 98, 1, 21, 4, 3, 0, 1, 1, 129, 1, 276, 1, 73, 10, 8, 1, 0, 1, 1, 257, 1, 794, 1, 273, 28, 30, 1, 5
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Comments

A(n,k) is the sum of k-th powers of proper divisors of n.

Examples

			Square array begins:
  0,  0,   0,   0,   0,    0,  ...
  1,  1,   1,   1,   1,    1,  ...
  1,  1,   1,   1,   1,    1,  ...
  2,  3,   5,   9,  17,   33,  ...
  1,  1,   1,   1,   1,    1,  ...
  3,  6,  14,  36,  98,  276,  ...

Links

Crossrefs

Columns k=0..5 give A032741, A001065, A067558, A276634, A279363, A279364.
Cf. A109974, A285425, A286880, A321259 (diagonal).

Programs

  • Mathematica
    Table[Function[k, DivisorSigma[k, n] - n^k][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j^k x^(2 j)/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

Formula

G.f. of column k: Sum_{j>=1} j^k*x^(2*j)/(1 - x^j).
Dirichlet g.f. of column k: zeta(s-k)*(zeta(s) - 1).
A(n,k) = 1 if n is prime.

A279364 Sum of 5th powers of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 33, 1, 276, 1, 1057, 244, 3158, 1, 9076, 1, 16840, 3369, 33825, 1, 67101, 1, 104182, 17051, 161084, 1, 290676, 3126, 371326, 59293, 555688, 1, 870552, 1, 1082401, 161295, 1419890, 19933, 2206525, 1, 2476132, 371537, 3336950, 1, 4646784, 1, 5315740, 821793, 6436376, 1, 9301876, 16808, 9868783
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 10 2016

Keywords

Examples

			a(10) = 1^5 + 2^5 + 5^5 = 3158, because 10 has 3 proper divisors {1,2,5}.
a(11) = 1^5 = 1, because 11 has 1 proper divisor {1}.

Links

Crossrefs

Cf. A000584, A001160, A013664.
Cf. A001065, A067558, A276634, A279363.

Programs

  • Mathematica
    Table[DivisorSigma[5, n] - n^5, {n, 50}]
  • PARI
    for(n=1, 50, print1(sigma(n, 5) - n^5,", ")) \\ Indranil Ghosh, Mar 18 2017
  • Python
    from sympy.ntheory import divisor_sigma
print([divisor_sigma(n,5) - n**5 for n in range(1,51)]) # Indranil Ghosh, Mar 18 2017

Formula

a(n) = 1 if n is prime.
a(p^k) = (p^(5*k) - 1)/(p^5 - 1) when p is prime.
Dirichlet g.f.: zeta(s-5)*(zeta(s) - 1).
a(n) = A001160(n) - A000584(n).
G.f.: -x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^6 + Sum_{k>=1} k^5 x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 18 2017
Sum_{k=1..n} a(k) ~ (zeta(6) - 1) * n^6 / 6. - Amiram Eldar, Jan 11 2025

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

A321259 a(n) = sigma_n(n) - n^n.

Original entry on oeis.org

0, 1, 1, 17, 1, 794, 1, 65793, 19684, 9766650, 1, 2194095090, 1, 678223089234, 30531927033, 281479271743489, 1, 150196195641350171, 1, 100000096466944316978, 558545874543637211, 81402749386839765307626, 1, 79501574308536809523296482, 298023223876953126
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 01 2018

Keywords

Comments

a(n) is the sum of n-th powers of proper divisors of n.

Links

Crossrefs

Cf. A000312, A001065, A023887, A067558, A276634, A279363, A279364, A292919, A321258.

Programs

  • Magma
    [DivisorSigma(n, n) - n^n: n in [1..30]]; // Vincenzo Librandi, Nov 02 2018
  • Mathematica
    Table[DivisorSigma[n, n] - n^n, {n, 25}]
nmax = 25; Rest[CoefficientList[Series[Sum[(k x)^(2 k)/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
  • PARI
    a(n) = sigma(n, n) - n^n; \\ Michel Marcus, Nov 02 2018

Formula

G.f.: Sum_{k>=1} (k*x)^(2*k)/(1 - (k*x)^k).
a(n) = A023887(n) - A000312(n).
a(n) = A321258(n,n).
a(n) = 1 if n is prime.
Showing 1-5 of 5 results.

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