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 1-4 of 4 results.

A346841 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^sin(x).

Original entry on oeis.org

1, 0, 2, 9, 40, 360, 2480, 28833, 266936, 3562920, 45634258, 659631225, 10231705196, 176661237948, 3080315922294, 59430009554685, 1217593208993232, 25766943601055184, 583245289316927058, 13861911731632256457, 343615639889119016556, 8925102256331257339140, 242399591002192962709230
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000203, A053529, A335626, A346547, A346941, A347774.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^sin(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, sigma(k)*x^k/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, x^k/(k*(1-x^k))))))

Formula

E.g.f.: exp( sin(x) * Sum_{k>=1} sigma(k)*x^k/k ).
E.g.f.: exp( sin(x) * Sum_{k>=1} x^k/(k*(1 - x^k)) ).

A347774 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^tan(x).

Original entry on oeis.org

1, 0, 2, 9, 52, 450, 3410, 41748, 415952, 5985144, 79468648, 1263309960, 20581146056, 375092849040, 7053697259856, 144054799315560, 3108398855786496, 70281839877041088, 1687564595412611520, 42264952015652902656, 1114043035100431983744, 30552235678578565203840
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000203, A346547, A346841, A346941.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^tan(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(tan(x)*sum(k=1, N, sigma(k)*x^k/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(tan(x)*sum(k=1, N, x^k/(k*(1-x^k))))))

Formula

E.g.f.: exp( tan(x) * Sum_{k>=1} sigma(k)*x^k/k ).
E.g.f.: exp( tan(x) * Sum_{k>=1} x^k/(k*(1 - x^k)) ).

A346940 Numbers whose square starts with exactly 4 identical digits.

Original entry on oeis.org

2357, 2582, 3334, 4714, 5774, 6667, 8165, 8819, 9428, 10542, 10543, 10544, 10545, 14907, 14908, 14909, 18257, 18258, 18259, 21081, 21082, 21083, 23570, 23571, 25819, 25820, 27888, 27889, 29813, 29814, 31622, 33332, 33333, 33335, 33336, 33337, 33338, 33339, 33340, 33341, 33342
Offset: 1

Views

Author

Bernard Schott, Aug 08 2021

Keywords

Comments

If m is a term, 10*m is another term.
Differs from A132391 where only at least 4 identical digits are required; indeed, 10541 is the first term of A132391 that is not in this sequence (see Example section), the next one is 33346.

Examples

			2357 is a term because 2357^2 = 5555449 starts with four 5's.
10541 is not a term because 10541^2 = 111112681 starts with five 1's.

Crossrefs

Cf. A346941, A346942.
Supersequences: A131573, A132391.
Similar with: A346812 (2 digits), A346891 (3 digits).

Programs

  • Mathematica
    q[n_] := SameQ @@ (d = IntegerDigits[n^2])[[1 ;; 4]] && d[[5]] != d[[1]]; Select[Range[100, 33350], q] (* Amiram Eldar, Aug 08 2021 *)
  • Python
    def ok(n):
    s = str(n*n)
    return len(s) > 4 and s[0] == s[1] == s[2] == s[3] != s[4]
print(list(filter(ok, range(33343)))) # Michael S. Branicky, Aug 08 2021

A347893 E.g.f.: Product_{k>=1} (1 + x^k)^cos(x).

Original entry on oeis.org

1, 1, 2, 9, 30, 195, 1545, 12474, 95564, 1199397, 14287845, 167518846, 2341450386, 34489552331, 540927170147, 10114629115798, 175935142966408, 3184271322683385, 68623817313870153, 1442553498798565142, 31856896467060026670, 787164874800260366287, 19097783293834170329239
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A265024, A346941, A347817, A347894, A347899.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^cos(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(cos(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))

Formula

E.g.f.: exp( cos(x) * Sum_{k>=1} x^k / (k*(1 - x^(2*k))) ). - Ilya Gutkovskiy, Sep 18 2021
E.g.f.: exp( cos(x) * Sum_{k>=1} A000593(k)*x^k/k ). - Seiichi Manyama, Sep 18 2021
Showing 1-4 of 4 results.

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

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