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.

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A075763 Numbers k that divide A001003(k-1).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 261, 263, 269
Offset: 1

Views

Author

Benoit Cloitre, Oct 09 2002

Keywords

Comments

All the odd primes are in the sequence.

Links

Crossrefs

Cf. A001003, A075764 (composite terms).

Programs

  • Mathematica
    s = {}; k1 = k2 = 1; Do[k3 = ((6*n - 9)*k2 - (n - 3)*k1)/n; If[Divisible[k3, n], AppendTo[s, n]]; k1 = k2; k2 = k3, {n, 3, 300}]; s (* Amiram Eldar, Jun 28 2022 *)

A075764 Schroeder pseudoprimes: Composites k that divide the k-th Schroeder number A001003(k-1).

Original entry on oeis.org

105, 261, 301, 693, 1605, 1755, 2151, 2905, 2907, 3393, 3875, 4641, 4833, 5005, 5655, 6279, 6913, 7161, 8883, 9405, 10899, 11025, 11289, 15687, 17199, 19203, 22275, 27387, 36855, 37791, 50007, 50463, 53493, 54891, 55209, 55755, 63327, 64337
Offset: 1

Views

Author

Benoit Cloitre, Oct 09 2002

Keywords

Examples

			105 is a term because A001003(105) = 15646506064359350392347086201481965698808674470977505246623827696393838448345 which is divisible by 105.
105 is a term because A001003(104) = 15646506064359350392347086201481965698808674470977505246623827696393838448345 which is divisible by 105.

Links

Crossrefs

Intersection of A002808 and A075763.
Cf. A001003, A013998, A075762.

Programs

  • Mathematica
    s = {}; k1 = k2 = 1; Do[k3 = ((6*n - 9)*k2 - (n - 3)*k1)/n; If[CompositeQ[n] && Divisible[k3, n], AppendTo[s, n]]; k1 = k2; k2 = k3, {n, 3, 10^5}]; s (* Amiram Eldar, Jun 28 2022 *)
  • PARI
    x1 = 1; x2 = 1; for (n = 3, 100000, x = (3*(2*n - 3)*x1 - (n - 3)*x2)/n; if (!isprime(n), if (!(x%n), print(n))); x2 = x1; x1 = x); \\ David Wasserman, Feb 23 2005

Extensions

More terms from David Wasserman, Feb 23 2005

A174810 A transform of the little Schroeder numbers A001003.

Original entry on oeis.org

1, 1, 4, 17, 81, 410, 2169, 11847, 66306, 378297, 2192011, 12864668, 76313865, 456837181, 2756271064, 16743326577, 102319639173, 628599899558, 3880049052441, 24051163355499, 149654739889478, 934426798835377
Offset: 0

Views

Author

Paul Barry, Mar 29 2010

Keywords

Comments

Hankel transform is A174811.

Links

Programs

  • Mathematica
    CoefficientList[Series[(1+x+x^2-Sqrt[1-6*x-5*x^2+2*x^3+x^4])/(4*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 30 2014 *)
  • PARI
    x='x+O('x^66); Vec((1+x+x^2-sqrt(1-6*x-5*x^2+2*x^3+x^4))/(4*x*(1+x))) \\ Joerg Arndt, Jan 30 2014

Formula

G.f.: (1+x+x^2-sqrt(1-6x-5x^2+2x^3+x^4))/(4x(1+x));
G.f.: 1/(1-x(1+x)/(1-2x(1+x)/(1-x(1+x)/(1-2x(1+x)/(1-... (continued fraction);
a(n)=sum{k=0..n, C(k,n-k)*A001003(k)}.
Recurrence: (n+1)*a(n) = (5-n)*a(n-5) - 3*(n-4)*a(n-4) + 3*(n-1)*a(n-3) + (11*n-13)*a(n-2) + (5*n-4)*a(n-1). - Fung Lam, Jan 30 2014

A227506 Schroeder triangle sums: a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1).

Original entry on oeis.org

1, 1, 7, 17, 121, 353, 2591, 8257, 61921, 207905, 1582791, 5501073, 42344121, 150827073, 1170747519, 4247388417, 33186295681, 122125206977, 959260792775, 3570473750929, 28167068630713, 105820555054241, 837838806587167, 3172136074486337
Offset: 1

Views

Author

Johannes W. Meijer, Jul 15 2013

Keywords

Comments

The terms of this sequence equal the Fi1 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

Crossrefs

Cf. A033877, A010683, A001003.

Programs

  • Maple
    A227506 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227506(n), n = 1..24); # Peter Luschny, Jul 17 2013
A227506 := proc(n): if type(n, odd) then A010683(n-1) else A010683(n-1) - A001003(n-1) fi: end: A010683 := proc(n): if n = 0 then 1 else (2/n)*add(binomial(n, k)* binomial(n+k+1, k-1), k=1..n) fi: end: A001003 := proc(n): if n = 0 then 1 else add(binomial(n, j)*binomial(n+j, n-1), j=0..n)/(2*n) fi: end: seq(A227506(n), n=1..24);
  • Mathematica
    T[n_, k_] := T[n, k] = Which[n == 1, 1, k < n, 0, True, T[n, k - 1] + T[n - 1, k - 1] + T[n - 1, k]];
a[n_] := Sum[T[2 k - 1, n], {k, 1, (n + 1)/2}];
Array[a, 24] (* Jean-François Alcover, Jul 11 2019, from Sage *)
  • Sage
    def A227506(n):
    @CachedFunction
    def T(n, k):
        if n==1: return 1
        if k A227506(n) for n in (1..24)]  # Peter Luschny, Jul 16 2013

Formula

a(n) = Sum_{k=1..floor((n+1)/2)} A033877(2*k-1,n).
a(2*n-1) = A010683(2*n-2) and a(2*n) = A010683(2*n-1) - A001003(2*n-1).
G.f.: (1-4*x+x^2 - sqrt(1-6*x+x^2) + x*sqrt(1+6*x+x^2))/(8*x).

A075762 A001003(p-1)/p where p runs through the odd primes.

Original entry on oeis.org

1, 9, 129, 47169, 1049913, 615504609, 15797864577, 11270258161281, 246696657053411097, 7087786816380876801, 178199489594187990894537, 158945391097530669380671857, 4791412124651983778003371329
Offset: 1

Views

Author

Benoit Cloitre, Oct 09 2002

Keywords

Formula

a(n)=A001003(prime(n+1)-1)/prime(n+1)

A101618 Indices of semiprimes in A001003.

Original entry on oeis.org

7, 11, 17, 19, 21, 239
Offset: 1

Views

Author

Jonathan Vos Post, Dec 09 2004

Keywords

Comments

No more terms < 2000. - David Wasserman, Mar 27 2008
a(7) >= 4377. A001003(4377) is a 3345-digit composite number with unknown factorization. Other possible terms are 4391, 45207, 45369, 45377, 131723, 131733, ... - Tyler Busby, Feb 08 2023

Examples

			a(1) = 7 because A001003(7) = 4279 = 11 * 389.
a(2) = 11 because A001003(11) = 2646723 = 3 * 882241.
a(3) = 17 because A001003(17) = 55909013009 = 41263 * 1354943.
a(4) = 19 because A001003(19) = 1618362158587 = 24413 * 66290999.
a(5) = 21 because A001003(21) = 47574827600981 = 3253 * 14624908577.

Links

  • Eric Weisstein's World of Mathematics, Semiprime.

Crossrefs

Cf. A001003, A001358.
Cf. A101619, A092839, A092840.

Formula

A001003(a(n)) is in A001358 (semiprimes).

Extensions

Edited by Franklin T. Adams-Watters, Nov 07 2006
Changed by Michael Somos, Mar 31 2007, because of the re-indexing of A001003.
a(6) from David Wasserman, Mar 27 2008

A101619 Semiprimes in A001003.

Original entry on oeis.org

4279, 2646723, 55909013009, 1618362158587, 47574827600981
Offset: 1

Views

Author

Jonathan Vos Post, Dec 09 2004

Keywords

Comments

Next term has 180 digits and is too large to include. - David Wasserman, Mar 27 2008

Links

  • Eric Weisstein's World of Mathematics, Semiprime.

Crossrefs

Cf. A001003, A001358.
Cf. A101618, A092839, A092840.

Formula

a(n) = A001003(A101618(n)). - David Wasserman, Mar 27 2008

Extensions

Edited by Franklin T. Adams-Watters, Nov 07 2006

A111993 Fifth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

Original entry on oeis.org

1, 5, 25, 125, 630, 3206, 16470, 85350, 445775, 2344595, 12408903, 66042795, 353259900, 1898119100, 10240583420, 55454182716, 301307002605, 1642192132625, 8975693643525, 49186242980105, 270186765784210
Offset: 0

Views

Author

Wolfdieter Lang, Sep 12 2005

Keywords

Links

Crossrefs

Cf. Fifth column of convolution triangle A011117. Fourth convolution: A010849.

Programs

  • Mathematica
    CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^5, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
  • PARI
    x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^5) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^5.
a(n)= (5/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+4,k-1), a(0)=1.
a(n) = 5*hypergeom([1-n, n+6], [2], -1), n>=1, a(0)=1.
Recurrence: n*(n+5)*a(n) = n*(7*n+23)*a(n-1) - (n+2)*(7*n-9)*a(n-2) + (n-3)*(n+2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 5*sqrt(3*sqrt(2)-4)*(17-12*sqrt(2)) * (3+2*sqrt(2))^(n+5)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

A111994 Sixth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

Original entry on oeis.org

1, 6, 33, 176, 930, 4908, 25954, 137712, 733539, 3922834, 21060099, 113481504, 613619332, 3328768344, 18112655748, 98833261600, 540705999621, 2965360687518, 16299708148901, 89784615643728, 495545294427558
Offset: 0

Views

Author

Wolfdieter Lang, Sep 12 2005

Keywords

Links

Crossrefs

Cf. Sixth column of convolution triangle A011117.

Programs

  • Mathematica
    CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^6, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
  • PARI
    x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^6) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^6.
a(n)= (6/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+5,k-1).
a(n) = 6*hypergeom([1-n, n+7], [2], -1), n>=1, a(0)=1.
Recurrence: n*(n+6)*a(n) = (7*n^2+30*n+5)*a(n-1) - (7*n^2+12*n-22)*a(n-2) + (n-3)*(n+3)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 3*sqrt(3*sqrt(2)-4)*(58-41*sqrt(2)) * (3+2*sqrt(2))^(n+6)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

A111995 Seventh convolution of Schroeder's (second problem) numbers A001003(n), n >= 0.

Original entry on oeis.org

1, 7, 42, 238, 1316, 7196, 39158, 212738, 1155889, 6287015, 34249404, 186920468, 1022134288, 5600420336, 30745867316, 169116129308, 931937277257, 5144687596447, 28449040406262, 157571572143538, 874089046798212
Offset: 0

Views

Author

Wolfdieter Lang, Sep 12 2005

Keywords

Links

Crossrefs

Cf. Seventh column of convolution triangle A011117.

Programs

  • Mathematica
    CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
  • PARI
    my(x='x+O('x^50)); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^7) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^7.
a(n) = (7/n)*Sum_{k=1..n} binomial(n,k)*binomial(n+k+6,k-1).
a(n) = 7*hypergeom([1-n, n+8], [2], -1), n >= 1, a(0)=1.
a(n) = fourth binomial transform of 1,3,2,6,4,12. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
Recurrence: n*(n+7)*a(n) = (7*n^2+37*n+12)*a(n-1) - (7*n^2+19*n-24)*a(n-2) + (n-3)*(n+4)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 7*sqrt(3*sqrt(2)-4)*(99-70*sqrt(2)) * (3+2*sqrt(2))^(n+7)/(32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

Extensions

Incorrect formula removed by Jason Yuen, Sep 07 2024
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