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.

A008969 Triangle of differences of reciprocals of unity.

Original entry on oeis.org

1, 1, 3, 1, 11, 7, 1, 50, 85, 15, 1, 274, 1660, 575, 31, 1, 1764, 48076, 46760, 3661, 63, 1, 13068, 1942416, 6998824, 1217776, 22631, 127, 1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255, 1, 1026576, 7245893376, 673781602752, 1413470290176, 117550462624, 747497920, 833375, 511
Offset: 1

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Author

N. J. A. Sloane

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  1,      3;
  1,     11,         7;
  1,     50,        85,         15;
  1,    274,      1660,        575,        31;
  1,   1764,     48076,      46760,      3661,       63;
  1,  13068,   1942416,    6998824,   1217776,    22631,    127;
  1, 109584, 104587344, 1744835904, 929081776, 30480800, 137845, 255;
  ...

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.

Links

Crossrefs

Cf. A001236, A001237, A001238, A001240, A001241, A001242.
Columns include A000254, A000424, A001236, A001237, A001238. Right-hand columns include A000225, A001240, A001241, A001242.

Programs

  • Maple
    T:= (n,k)-> `if`(k<=n, (n-k+2)!^k *
     add((-1)^(j+1)*binomial(n-k+2, j)/ j^k, j=1..n-k+2), 0):
seq(seq(T(n,k), k=0..n), n=0..7); # Alois P. Heinz, Sep 05 2008
  • Mathematica
    T[n_, k_] := If[k <= n, (n-k+2)!^k*Sum[(-1)^(j+1)*Binomial[n-k+2, j]/j^k, {j, 1, n-k+2}], 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

A001236 Differences of reciprocals of unity.

Original entry on oeis.org

15, 575, 46760, 6998824, 1744835904, 673781602752, 381495483224064, 303443622431870976, 327643295527342080000, 466962174913357393920000, 858175477913267353681920000, 1993920215002599923346309120000, 5758788816015998806424467537920000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
  • 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

Column 3 in triangle A008969.
Cf. A000254, A000424, A001237, A001238.

Programs

  • Maple
    a:= n-> (n+1)!^3* add((-1)^(k+1) *binomial(n+1, k)/ k^3, k=1..n+1):
seq (a(n), n=1..15);  # Alois P. Heinz, Sep 05 2008
  • Mathematica
    h = HarmonicNumber; a[n_] := ((n+1)!^3/6)*(h[n+1, 1]^3 + 3*h[n+1, 1]*h[n+1, 2] + 2*h[n+1, 3]); Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Feb 26 2015, after Vladeta Jovovic *)

Formula

a(n) = (n+1)!^3 * Sum_{i=1..n+1} Sum_{j=1..i} Sum_{k=1..j} 1/(i*j*k).
From Vladeta Jovovic, Jan 30 2005: (Start)
a(n) = (n!^3/6)*(H(n, 1)^3+3*H(n, 1)*H(n, 2)+2*H(n, 3)), where H(n, m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.
a(n) = (n!^3/6)*((Psi(n+1)+gamma)^3+3*(Psi(n+1)+gamma)*(-Psi(1, n+1)+1/6*Pi^2)+Psi(2, n+1)+2*Zeta(3)).
a(n) = n!^3*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^3.
Sum_{n>=0} a(n)*x^n/n!^3 = polylog(3, x/(x-1))/(x-1). (offset 2). (End)

Extensions

More terms from Alois P. Heinz, Sep 05 2008

A365009 Semiprimes that are the concatenation of two or more semiprimes.

Original entry on oeis.org

46, 49, 69, 94, 106, 146, 159, 214, 219, 226, 254, 259, 334, 339, 346, 386, 394, 415, 422, 446, 451, 458, 466, 469, 482, 485, 493, 514, 519, 554, 559, 579, 586, 589, 614, 622, 626, 629, 633, 634, 635, 649, 655, 662, 669, 674, 685, 687, 694, 695, 699, 746, 749, 779, 866, 869, 879, 914, 921, 922
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Aug 15 2023

Keywords

Comments

Conjecture: The fraction of semiprimes <= N that are in this sequence goes to 1 as N -> infinity. What is the first N for which that fraction >= 1/2?

Examples

			a(3) = 69 is a term because 69 = 3 * 23 is a semiprime and is the concatenation of the semiprimes 6 = 2 * 3 and 9 = 3 * 3.

Links

Crossrefs

Cf. A001358, A001238, A019549. Contains A107342.

Programs

  • Maple
    filter:= proc(n) local d,v;
  if numtheory:-bigomega(n) <> 2 then return false fi;
  for d from 1 to length(n)-1 do
     v:= n  mod 10^d;
     if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and g((n-v)/10^d) then return true fi
  od;
  false
end proc:
g:= proc(n) local d,v; option remember;
  if numtheory:-bigomega(n) = 2 then return true fi;
  for d from 1 to length(n)-1 do
    v:= n mod 10^d;
    if v >= 10^(d-1) and numtheory:-bigomega(v)=2 and procname((n-v)/10^d) then return true fi
  od;
  false
end proc:
select(filter, [$10..1000]);
Showing 1-3 of 3 results.

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

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