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 1-8 of 8 results.

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.

A034677 Sum of cubes of unitary divisors of n.

Original entry on oeis.org

1, 9, 28, 65, 126, 252, 344, 513, 730, 1134, 1332, 1820, 2198, 3096, 3528, 4097, 4914, 6570, 6860, 8190, 9632, 11988, 12168, 14364, 15626, 19782, 19684, 22360, 24390, 31752, 29792, 32769, 37296, 44226, 43344, 47450, 50654, 61740, 61544, 64638, 68922, 86688, 79508
Offset: 1

Views

Author

Erich Friedman

Keywords

Comments

A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.

Examples

			The unitary divisors of 6 are 1, 2, 3 and 6, so a(6) = 252.

Links

Crossrefs

Cf. A034444, A034448.
Row n=3 of A286880.

Programs

  • Mathematica
    scud[n_]:=Total[Select[Divisors[n],CoprimeQ[#,n/#]&]^3]; Array[scud,40] (* Harvey P. Dale, Oct 16 2016 *)
f[p_, e_] := p^(3*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)
  • PARI
    A034677_vec(len)={
        a000012=direuler(p=2,len, 1/(1-X)) ;
        a000578=direuler(p=2,len, 1/(1-p^3*X)) ;
        a000578x=direuler(p=2,len, 1-p^3*X^2) ;
        dirmul(dirmul(a000012,a000578),a000578x)
}
A034677_vec(70) /* via D.g.f., R. J. Mathar, Mar 05 2011 */

Formula

Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2s-3). - R. J. Mathar, Mar 04 2011
If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(3*k_j)). - Ilya Gutkovskiy, Nov 04 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (360 * Zeta(5)). - Vaclav Kotesovec, Feb 01 2019

A034678 Sum of fourth powers of unitary divisors.

Original entry on oeis.org

1, 17, 82, 257, 626, 1394, 2402, 4097, 6562, 10642, 14642, 21074, 28562, 40834, 51332, 65537, 83522, 111554, 130322, 160882, 196964, 248914, 279842, 335954, 390626, 485554, 531442, 617314, 707282, 872644, 923522, 1048577, 1200644
Offset: 1

Views

Author

Erich Friedman

Keywords

Links

Crossrefs

Cf. A034444, A034448.
Row n=4 of A286880.

Programs

  • Mathematica
    Table[Total[Select[Divisors[n], CoprimeQ[#, n/#] &]^4], {n, 1, 50}] (* Vaclav Kotesovec, Feb 01 2019 *)
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(4*Last[#]) & /@ FactorInteger[n]); s = Array[a, 50] (* Amiram Eldar, Aug 10 2019 *)
  • PARI
    A000012=direuler(p=2,119, 1/(1-X)) ;
A000583=direuler(p=2,119, 1/(1-p^4*X)) ;
A000290x=direuler(p=2,119, 1-p^4*X^2) ;
dirmul(dirmul(A000012,A000583),A000290x) /* R. J. Mathar, Mar 05 2011 */

Formula

Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(2*s-4). - R. J. Mathar, Mar 04 2011
If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(4*k_j)). - Ilya Gutkovskiy, Nov 04 2018
Sum_{k=1..n} a(k) ~ 189 * Zeta(5) * n^5 / Pi^6. - Vaclav Kotesovec, Feb 01 2019

A034682 Sum of eighth powers of unitary divisors.

Original entry on oeis.org

1, 257, 6562, 65537, 390626, 1686434, 5764802, 16777217, 43046722, 100390882, 214358882, 430053794, 815730722, 1481554114, 2563287812, 4294967297, 6975757442, 11063007554, 16983563042, 25600456162, 37828630724
Offset: 1

Views

Author

Erich Friedman

Keywords

Links

Crossrefs

Cf. A034444, A034448.
Row n=8 of A286880.

Programs

  • Mathematica
    Table[Total[Select[Divisors[n], CoprimeQ[#, n/#] &]^8], {n, 1, 50}] (* Vaclav Kotesovec, Feb 07 2019 *)
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(8*Last[#]) & /@ FactorInteger[n]); s = Array[a, 50] (* Amiram Eldar, Aug 10 2019 *)

Formula

Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(2s-8). - R. J. Mathar, Apr 12 2011
If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(8*k_j)). - Ilya Gutkovskiy, Nov 04 2018
Sum_{k=1..n} a(k) ~ 10395*Zeta(9)*n^9 / Pi^10. - Vaclav Kotesovec, Feb 07 2019

A034679 Sum of fifth powers of unitary divisors.

Original entry on oeis.org

1, 33, 244, 1025, 3126, 8052, 16808, 32769, 59050, 103158, 161052, 250100, 371294, 554664, 762744, 1048577, 1419858, 1948650, 2476100, 3204150, 4101152, 5314716, 6436344, 7995636, 9765626, 12252702, 14348908, 17228200, 20511150, 25170552
Offset: 1

Views

Author

Erich Friedman

Keywords

Links

Crossrefs

Cf. A034444, A034448.
Row n=5 of A286880.

Programs

  • Mathematica
    Table[Total[Select[Divisors[n], CoprimeQ[#, n/#] &]^5], {n, 1, 50}] (* Vaclav Kotesovec, Feb 07 2019 *)
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(5*Last[#]) & /@ FactorInteger[n]); s = Array[a, 50] (* Amiram Eldar, Aug 10 2019 *)

Formula

Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(2s-5). - R. J. Mathar, Apr 12 2011
If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(5*k_j)). - Ilya Gutkovskiy, Nov 04 2018
Sum_{k=1..n} a(k) ~ (Pi*n)^6 / (5670*Zeta(7)). - Vaclav Kotesovec, Feb 07 2019

A034680 Sum of sixth powers of unitary divisors.

Original entry on oeis.org

1, 65, 730, 4097, 15626, 47450, 117650, 262145, 531442, 1015690, 1771562, 2990810, 4826810, 7647250, 11406980, 16777217, 24137570, 34543730, 47045882, 64019722, 85884500, 115151530, 148035890, 191365850, 244140626, 313742650
Offset: 1

Views

Author

Erich Friedman

Keywords

Links

Crossrefs

Cf. A034444, A034448.
Row n=6 of A286880.

Programs

  • Mathematica
    Total[#^6]&/@Table[Select[Divisors[n],GCD[#,n/#]==1&],{n,30}] (* Harvey P. Dale, Jul 17 2011 *)
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(6*Last[#]) & /@ FactorInteger[n]); s = Array[a, 50] (* Amiram Eldar, Aug 10 2019 *)

Formula

Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(2s-6). - R. J. Mathar, Apr 12 2011
If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(6*k_j)). - Ilya Gutkovskiy, Nov 04 2018
Sum_{k=1..n} a(k) ~ 1350*Zeta(7)*n^7 / Pi^8. - Vaclav Kotesovec, Feb 07 2019

A034681 Sum of seventh powers of unitary divisors.

Original entry on oeis.org

1, 129, 2188, 16385, 78126, 282252, 823544, 2097153, 4782970, 10078254, 19487172, 35850380, 62748518, 106237176, 170939688, 268435457, 410338674, 617003130, 893871740, 1280094510, 1801914272, 2513845188, 3404825448
Offset: 1

Views

Author

Erich Friedman

Keywords

Links

Crossrefs

Cf. A034444, A034448.
Row n=7 of A286880.

Programs

  • Mathematica
    Table[Total[Select[Divisors[n], CoprimeQ[#, n/#] &]^7], {n, 1, 50}] (* Vaclav Kotesovec, Feb 07 2019 *)
a[1] = 1; a[n_] := Times @@ (1 + First[#]^(7*Last[#]) & /@ FactorInteger[n]); s = Array[a, 50] (* Amiram Eldar, Aug 10 2019 *)

Formula

Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(2s-7). - R. J. Mathar, Apr 12 2011
If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(7*k_j)). - Ilya Gutkovskiy, Nov 04 2018
Sum_{k=1..n} a(k) ~ (Pi*n)^8 / (75600*Zeta(9)). - Vaclav Kotesovec, Feb 07 2019

A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 9, 2, 1, 0, 16, 27, 4, 5, 2, 0, 32, 81, 8, 25, 5, 1, 0, 64, 243, 16, 125, 13, 7, 1, 0, 128, 729, 32, 625, 35, 49, 2, 1, 0, 256, 2187, 64, 3125, 97, 343, 4, 3, 2, 0, 512, 6561, 128, 15625, 275, 2401, 8, 9, 7, 1, 0, 1024, 19683, 256, 78125, 793, 16807, 16, 27, 29, 11, 2
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2018

Keywords

Examples

			Square array begins:
  0,  0,   0,    0,    0,     0,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  3,   9,   27,   81,   243,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  5,  25,  125,  625,  3125,  ...
  2,  5,  13,   35,   97,   275,  ...

Links

Crossrefs

Columns k=0..4 give A001221, A008472, A005063, A005064, A005065.
Cf. A109974, A200768 (diagonal), A285425, A286880, A321258.

Programs

  • Mathematica
    Table[Function[k, Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[Prime[j]^k x^Prime[j]/(1 - x^Prime[j]), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
  • PARI
    T(n,k)={vecsum([p^k | p<-factor(n)[,1]])}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

Formula

G.f. of column k: Sum_{j>=1} prime(j)^k*x^prime(j)/(1 - x^prime(j)).
Showing 1-8 of 8 results.

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

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