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 1-3 of 3 results.

A002443 Numerator in Feinler's formula for unsigned Bernoulli number |B_{2n}|.

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 1382, 420, 10851, 438670, 7333662, 51270780, 7090922730, 2155381956, 94997844116, 68926730208040, 1780853160521127, 541314450257070, 52630543106106954746, 15997766769574912140, 10965474176850863126142, 1003264444985926729776060, 35069919669919290536128980
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A002443/A002444 = |B_{2n}| (see also A000367/A002445).
a(n) is a nontrivial multiple of A000367(n) if gcd(a(n),A002444(n)) > 1. Furthermore, all terms here are positive, whereas the terms of A000367 retain the sign of B_{2n}, e.g., a(8)/A002444(8) = 10851/1530 is the absolute value of A000367(8)/A002445(8) = -3617/510 = B_{16}. - M. F. Hasler, Jan 05 2016

References

  • H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 208.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002444, A000367, A002445, A266742, A266743, A266911.

Formula

See Davis, Vol. 2, p. 206, second displayed equation, where a(n) appears as c_{2k}. Note that the recurrence for c_{2k} involves an extra term c_1 = 1 (which is not a term of the present sequence), and also the numbers M_i^{2k} given in A266743. However, given that contemporary Computer Algebra Systems can easily calculate Bernoulli numbers, and A002444 has a simple formula, the best way to compute a(n) today is via a(n) = A002444(n)*|B_{2n}|. - N. J. A. Sloane, Jan 08 2016

Extensions

Name amended following a suggestion from T. D. Noe. - M. F. Hasler, Jan 05 2016
Edited with new definition, further terms, and scan of source by N. J. A. Sloane, Jan 08 2016

A002444 Denominator in Feinler's formula for unsigned Bernoulli number |B_{2n}|.

Original entry on oeis.org

1, 6, 30, 84, 90, 132, 5460, 360, 1530, 7980, 13860, 8280, 81900, 1512, 3480, 114576, 117810, 1260, 3838380, 32760, 568260, 1191960, 869400, 236880, 9746100, 525096, 629640, 351120, 198360, 42480, 1362881520, 4324320, 1093950, 33008220, 434700, 843480, 46233287100, 102702600, 1081080
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A002443/A002444 = |B_{2n}| (see also A000367/A002445).

References

  • H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 208.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002443, A000367, A002445, A266742, A266743, A266911.

Programs

  • Maple
    with(numtheory);
g:=proc(m) local i,n; n:=2*m;
mul(ithprime(i)^floor(n/(ithprime(i)-1)),i=1..pi(n+1));
%/n!;
end;
[seq(g(m),m=0..40)]; # N. J. A. Sloane, Jan 08 2016
  • Mathematica
    a[n_] := Product[Prime[i]^Floor[2n/(Prime[i]-1)], {i, 1, PrimePi[2n+1]}]/(2n)!;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 08 2023 *)

Formula

Let p_i denote the i-th prime, and let V(n,i) = floor(n/(prime(i)-1)) = A266742(n,i).
Then a(n) = (Prod_i (p_i)^V(n,i))/n!.
(See Davis, Vol. 2, p. 206, first displayed equation, where a(n) appears as d_{2k}.)

Extensions

Name amended upon suggestion by T. D. Noe, by M. F. Hasler, Jan 05 2016
Edited with new definition, more terms, and scan of source by N. J. A. Sloane, Jan 08 2016

A266742 Irregular triangle read by rows: T(n,k) = floor(n/(prime(k)-1)), n>=1, 1 <= k <= pi(n+1), where pi is A000720.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 1, 5, 2, 1, 6, 3, 1, 1, 7, 3, 1, 1, 8, 4, 2, 1, 9, 4, 2, 1, 10, 5, 2, 1, 1, 11, 5, 2, 1, 1, 12, 6, 3, 2, 1, 1, 13, 6, 3, 2, 1, 1, 14, 7, 3, 2, 1, 1, 15, 7, 3, 2, 1, 1, 16, 8, 4, 2, 1, 1, 1, 17, 8, 4, 2, 1, 1, 1, 18, 9, 4, 3, 1, 1, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Jan 08 2016

Keywords

Examples

			Triangle begins:
[1]
[2, 1]
[3, 1]
[4, 2, 1]
[5, 2, 1]
[6, 3, 1, 1]
[7, 3, 1, 1]
[8, 4, 2, 1]
[9, 4, 2, 1]
[10, 5, 2, 1, 1]
[11, 5, 2, 1, 1]
[12, 6, 3, 2, 1, 1]
[13, 6, 3, 2, 1, 1]
[14, 7, 3, 2, 1, 1]
...

Links

  • H. T. Davis, Tables of the Mathematical Functions, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.] See Table 1 on page 205.

Crossrefs

Cf. A000720, A002443, A002444, A266743.

Programs

  • Maple
    with(numtheory);
f:=n->[seq(floor(n/(ithprime(i)-1)),i=1..pi(n+1))];
for n from 1 to 20 do lprint(f(n)); od:
Showing 1-3 of 3 results.

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

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