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.

Previous Showing 11-13 of 13 results.

A120052 Number of 11-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 0, 7, 138, 1878, 23448, 279286, 3230577, 36585097, 407818620, 4490844534, 48972151631, 529781669333, 5693047157230, 60832290450373, 646862625625663, 6849459596884350, 72259172519243461
Offset: 0

Views

Author

Robert G. Wilson v, Feb 07 2006

Keywords

Examples

			There are 7 eleven-almost primes up to 10000: 2048, 3072, 4608, 5120, 6912, 7168, and 7680.

Crossrefs

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[11, 10^n], {n, 12}]

Extensions

a(14) from Robert G. Wilson v, Jan 07 2007
a(15)-a(19) from Henri Lifchitz, Mar 18 2025

A116385 Expansion of e.g.f. Bessel_I(2,2x) + 2*Bessel_I(3,2x) + Bessel_I(4,2x).

Original entry on oeis.org

0, 0, 1, 2, 5, 10, 21, 42, 84, 168, 330, 660, 1287, 2574, 5005, 10010, 19448, 38896, 75582, 151164, 293930, 587860, 1144066, 2288132, 4457400, 8914800, 17383860, 34767720, 67863915, 135727830, 265182525, 530365050, 1037158320, 2074316640
Offset: 0

Views

Author

Paul Barry, Feb 12 2006

Keywords

Comments

Third column of the Riordan array A116382.
Apart from its root term -1: central terms of the triangle in A051631: a(n) = A051631(n+1, [(n+1)/2]). - Reinhard Zumkeller, Nov 13 2011

Links

Crossrefs

Cf. A001405.

Programs

  • Haskell
    a116385 n = a051631 (n+1) $ (n+1) `div` 2
-- Reinhard Zumkeller, Nov 13 2011
  • Mathematica
    With[{nn=40},CoefficientList[Series[BesselI[2,2x]+2BesselI[3,2x]+ BesselI[ 4,2x],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Sep 14 2011 *)
  • PARI
    a(n)= binomial(n+3, (n+3)\2) - 3*binomial(n+1, (n+1)\2) \\ Bill McEachen, Dec 12 2022

Formula

E.g.f.: (d/dx)(Bessel_I(3,2x),x) + 2*Bessel_I(3,2x).
a(n) = C(n+1,floor((n-2)/2))*(1+(-1)^n)/2 + C(n,floor((n-3)/2))*(1-(-1)^n).
Conjecture: (n+4)*a(n) -2*a(n-1) +(-7*n-8)*a(n-2) +6*a(n-3) +12*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 13 2014
a(n) = A001405(n+3) - 3*A001405(n+1) (from Eremin link). - Bill McEachen, Dec 12 2022
G.f.: (-1 - x + x^2 + B(x) - 3*x^2*B(x))/x^3, where B(x) is the g.f. of A001405. - Gennady Eremin, Oct 09 2023

A159965 Riordan array (1/sqrt(1-4x), (1-2x-(1-3x)c(x))/(x*sqrt(1-4x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 20, 21, 8, 1, 70, 84, 45, 11, 1, 252, 330, 220, 78, 14, 1, 924, 1287, 1001, 455, 120, 17, 1, 3432, 5005, 4368, 2380, 816, 171, 20, 1, 12870, 19448, 18564, 11628, 4845, 1330, 231, 23, 1, 48620, 75582, 77520, 54264, 26334, 8855, 2024, 300, 26, 1
Offset: 0

Views

Author

Paul Barry, Apr 28 2009

Keywords

Comments

Product of A007318 and A114422. Product of A007318^2 and A116382. Row sums are A108080.
Diagonal sums are A108081.
Riordan array (1/sqrt(1 - 4*x), x*c(x)^3) obtained from A092392 by taking every third column starting from column 0; x*c(x)^3 is the o.g.f. for A000245. - Peter Bala, Nov 24 2015

Examples

			Triangle begins
1,
2, 1,
6, 5, 1,
20, 21, 8, 1,
70, 84, 45, 11, 1,
252, 330, 220, 78, 14, 1,
924, 1287, 1001, 455, 120, 17, 1,
3432, 5005, 4368, 2380, 816, 171, 20, 1

Links

Crossrefs

Cf. A000245, A007318, A092392, A108080, A108081, A114422, A116382.

Programs

  • Magma
    /* As triangle */ [[Binomial(2*n+k, n+2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 27 2015

Formula

Number triangle T(n,k) = Sum_{j = 0..n} binomial(n+k,j-k)*binomialC(n,j).
T(n,k) = binomial(2*n + k, n + 2*k). - Peter Bala, Nov 24 2015
Previous Showing 11-13 of 13 results.

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