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 1-6 of 6 results.

A357267 Expansion of e.g.f. -LambertW(x * (1 - exp(x))).

Original entry on oeis.org

0, 0, 2, 3, 28, 125, 1506, 12607, 186600, 2352681, 41839750, 705821171, 14818593516, 311784460429, 7603945309338, 190868446707135, 5328147004384336, 154893585657590609, 4884408906341245326, 161057122218190660555, 5671407469802947722900
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2022

Keywords

Links

Crossrefs

Cf. A048802, A355843, A357265.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-lambertw(x*(1-exp(x))))))
  • PARI
    a(n) = n!*sum(k=1, n\2, k^(k-1)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=1..floor(n/2)} k^(k-1) * Stirling2(n-k,k)/(n-k)!.

A355874 Expansion of e.g.f. -LambertW(x^2 * log(1-x))/2.

Original entry on oeis.org

0, 0, 0, 3, 6, 20, 450, 3024, 21840, 449280, 5690160, 68579280, 1491462720, 27798076800, 485405784864, 11821894207200, 285057334598400, 6578025489584640, 183420564173141760, 5342163886869062400, 152988752430721267200, 4897735504358795965440
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A052807, A355993, A357265.
Cf. A355179.

Programs

  • Mathematica
    With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^2 * Log[1 - x]]/2, {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2*log(1-x)))/2))
  • PARI
    a(n) = n!*sum(k=1, n\3, k^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/2;

Formula

a(n) = (n!/2) * Sum_{k=1..floor(n/3)} k^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

A355993 Expansion of e.g.f. -LambertW(x^3 * log(1-x))/6.

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 40, 210, 8064, 70560, 640800, 6375600, 189383040, 3165402240, 48879754560, 762766804800, 21652937349120, 525738717504000, 11796584629939200, 259139188966694400, 7842638783736115200, 240231375437935795200, 7066934411387842252800
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A052807, A355874, A357265.

Programs

  • Mathematica
    With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^3 * Log[1 - x]]/6, {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3*log(1-x)))/6))
  • PARI
    a(n) = n!*sum(k=1, n\4, k^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!)/6;

Formula

a(n) = (n!/6) * Sum_{k=1..floor(n/4)} k^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.

A355994 Expansion of e.g.f. -LambertW(x^2/2 * log(1-x)).

Original entry on oeis.org

0, 0, 0, 3, 6, 20, 270, 1764, 12600, 169560, 1937880, 22300740, 349806600, 5556245760, 89073856872, 1678920566400, 33550354656000, 687175528253760, 15462823882213440, 370285712520237360, 9180722384533375200, 242398467521271149760
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A052807, A355995, A357265.

Programs

  • Mathematica
    With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^2/2 * Log[1 - x]], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2/2*log(1-x)))))
  • PARI
    a(n) = n!*sum(k=1, n\3, k^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/3)} k^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).

A355995 Expansion of e.g.f. -LambertW(x^3/6 * log(1-x)).

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 40, 210, 2464, 20160, 178800, 1755600, 22323840, 289729440, 3950069760, 57127870800, 921032555520, 15786602832000, 284810759251200, 5394363163862400, 108742028591923200, 2312415679065811200, 51543520889668684800, 1199641884471310156800
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A052807, A355994, A357265.

Programs

  • Mathematica
    With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^3/6 * Log[1 - x]], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3/6*log(1-x)))))
  • PARI
    a(n) = n!*sum(k=1, n\4, k^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/4)} k^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).

A367266 Numbers k such that there exists i >= 1 such that k divides 3^3^i + 1.

Original entry on oeis.org

1, 2, 4, 7, 14, 19, 28, 37, 38, 74, 76, 133, 148, 163, 259, 266, 326, 487, 518, 532, 652, 703, 974, 1036, 1141, 1297, 1406, 1459, 1948, 2282, 2594, 2812, 2917, 2918, 3097, 3409, 4564, 4921, 5188, 5834, 5836, 6031, 6194, 6818, 9079, 9253, 9842, 10213, 11668, 12062, 12388, 13636, 18019, 18158, 18506, 19441, 19684, 19927
Offset: 1

Views

Author

Jianing Song, Nov 11 2023

Keywords

Comments

Note that 3^3^i + 1 divides 3^3^(i+1) + 1, so this sequence is also numbers k such that k divides 3^3^i + 1 for all sufficiently large i.
Also numbers k such that there exists i >= 1 such that k divides 3^^i + 1, where 3^^i = 3^3^...^3 (i times) = A014220(i-1).
Write k = 2^{e_0} * Product_{j=1..r} (p_j)^(e_j), then k is a term if and only if e_0 <= 2, and ord(3,(p_j)^(e_j)) is 2 times a power of 3 for every 1 <= j <= r, where ord(a,k) is the multiplicative order of a modulo k: 3^3^i == -1 (mod k) if and only if 3^3^i == -1 (mod 2^{e_0}), and 3^3^i == -1 (mod (p_j)^(e_j)) for every 1 <= j <= r. This is in turn equivalent to e_0 <= 2, and ord(3,(p_j)^(e_j)) being even, and 3^i == ord(3,(p_j)^(e_j))/2 (mod ord(3,(p_j)^(e_j))) for every 1 <= j <= r. As a result, such i exists if and only if e_0 <= 2, and ord(3,(p_j)^(e_j)) is 2 times a power of 3 for every 1 <= j <= r. In other words, each term is a product of a number in {1,2,4} and odd prime powers q such that ord(3,q) is 2 times a power of 3.
If an term k is not squarefree, then it is divisible by p^2, where p is a Wieferich prime to base 3 (A014127) such that ord(3,p) is 2 times a power of 3. No such p is known.

Examples

			Suppose that q is an odd prime power such that ord(3,q) = 2*3^e. e = 1 gives q = 7; e = 2 gives q = 19; e = 3 gives q = 19441 and q = 19927; e = 4 gives q = 163, 1297, 208657, 224209, 5879415781.

Crossrefs

Cf. A094358 (squarefree divisors of 2^2^i - 1), A357265 (divisors of 3^3^i - 1), A014127.
The subsequence of primes is given by A367649.

Programs

  • PARI
    isA357266_primepower(q) = if(q%2==0, q%8!=0, my(d); (q%3!=0) && ((d=znorder(Mod(3,q)))%2==0) && isprimepower(3*d/2))
isA357266(n) = my(f=factor(n)); for(i=1, #f~, if(!isA357266_primepower(f[i,1]^f[i,2]), return(0))); return(1)
Showing 1-6 of 6 results.

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

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