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.

User: Roberto E. Martinez II

Roberto E. Martinez II's wiki page.

Roberto E. Martinez II has authored 9 sequences.

A181629 Positive integers k = p_1^{r_1} ... p_n^{r_n} such that sum_{i=1..n} p_i^{-r_i} >= 1 (Non-Hyperbolic Integers).

Original entry on oeis.org

1, 30, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1050, 1110, 1218, 1230, 1290, 1302, 1410, 1470, 1554, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2370, 2490, 2670, 2730, 2910, 3030, 3090, 3210, 3270, 3390, 3570, 3630, 3810
Offset: 1

Views

Author

Roberto E. Martinez II, Nov 02 2010, Nov 05 2010

Keywords

Comments

First odd term greater than 1 is 3234846615. - Robert G. Wilson v, Nov 04 2010
Also numbers n such that A028236(n)/n >= 1. - Klaus Brockhaus, Nov 06 2010

Examples

			a(2) = 30, since 30 = 2*3*5 and 1/2 + 1/3 + 1/5 = 31/30 >= 1.

Crossrefs

Cf. A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j). - Klaus Brockhaus, Nov 06 2010

Programs

  • Magma
    [1] cat [ k: k in [2..4000] | &+[ f[i, 1]^-f[i, 2]: i in [1..#f] ] ge 1 where f is Factorization(k) ]; // Klaus Brockhaus, Nov 06 2010
  • Mathematica
    DeleteCases[ Table[k; A = FactorInteger[k]; If[Sum[1/A[[j]][[1]]^A[[j]][[2]], {j, 1, Length[A]}] >= 1, k, 0], {k, 1, 3900}], 0]
fQ[n_] := Block[{fi = Transpose@ FactorInteger@ n}, Plus @@ (1/(First@fi ^ Last@fi)) >= 1]; Select[Range@ 3900, fQ] (* Robert G. Wilson v, Nov 04 2010 *)

A066655 Number of partitions of n*(n-1)/2.

Original entry on oeis.org

1, 1, 3, 11, 42, 176, 792, 3718, 17977, 89134, 451276, 2323520, 12132164, 64112359, 342325709, 1844349560, 10015581680, 54770336324, 301384802048, 1667727404093, 9275102575355, 51820051838712, 290726957916112, 1637293969337171, 9253082936723602
Offset: 1

Views

Author

Roberto E. Martinez II, Jan 10 2002

Keywords

Comments

Number of partitions of the number of edges of the complete graph of order n, K_n.

Examples

			a(4) = p(6) = 11.

Links

Crossrefs

Cf. A000041, A000217, A007294, A104383.
Cf. A173519. - Reinhard Zumkeller, Feb 20 2010

Programs

  • Mathematica
    Table[PartitionsP[n(n-1)/2], {n, 1, 30}]
  • MuPAD
    combinat::partitions::count(binomial(n+2,n)) $n=-1..40 // Zerinvary Lajos, Apr 16 2007
  • PARI
    a(n) = numbpart(n*(n-1)/2); \\ Michel Marcus, Dec 18 2017

Formula

a(n) = p(n*(n-1)/2) = A000041(n*(n-1)/2).
a(n) ~ exp(Pi*sqrt(n*(n-1)/3))/(2*sqrt(3)*n*(n - 1)). - Ilya Gutkovskiy, Jan 13 2017
a(n) ~ exp(Pi*(n - 1/2) / sqrt(3)) / (2*sqrt(3)*n^2). - Vaclav Kotesovec, May 17 2018

Extensions

More terms from Vladeta Jovovic, Jan 12 2002
Edited by Dean Hickerson, Jan 14 2002

A066543 Number of spanning trees in the line graph of the product of two cycle graphs, each of order n, L(C_n x C_n).

Original entry on oeis.org

782757789696, 5976745079881894723584, 29514790517935282585600000000000000, 95296975201657487970461602120230307486331043840000, 202142993853936783750487849288950496428731602354031286611374533246976
Offset: 3

Views

Author

Roberto E. Martinez II, Jan 07 2002

Keywords

Examples

			NumberOfSpanningTrees(L(C_3 x C_3)) = 782757789696

Crossrefs

Cf. A212800.

Programs

  • Mathematica
    NumberOfSpanningTrees[LineGraph[GraphProduct[Cycle[n], Cycle[n]]]] (* First load package DiscreteMath`Combinatorica` *)

Formula

a(n) = 2^(3*n^2-1) * A212800(n). - Sean A. Irvine, Oct 25 2023

Extensions

Edited by Dean Hickerson, Jan 14 2002
a(7) from Sean A. Irvine, Oct 25 2023

A064761 a(n) = 15*n^2.

Original entry on oeis.org

0, 15, 60, 135, 240, 375, 540, 735, 960, 1215, 1500, 1815, 2160, 2535, 2940, 3375, 3840, 4335, 4860, 5415, 6000, 6615, 7260, 7935, 8640, 9375, 10140, 10935, 11760, 12615, 13500, 14415, 15360, 16335, 17340, 18375, 19440, 20535, 21660, 22815
Offset: 0

Views

Author

Roberto E. Martinez II, Oct 18 2001

Keywords

Comments

Number of edges in a complete 6-partite graph of order 6n, K_n,n,n,n,n,n.

Links

Crossrefs

Cf. A000217, A000290, A008585, A008587, A022272, A033428, A033429, A033581, A033583.
Cf. A008597, A195046.

Programs

Formula

a(n) = 15*A000290(n) = 5*A033428(n) = 3*A033429(n). - Omar E. Pol, Dec 13 2008
a(n) = A008587(n)*A008585(n). - Reinhard Zumkeller, Apr 12 2010
a(n) = a(n-1) + 30*n - 15 for n > 0, a(0)=0. - Vincenzo Librandi, Dec 15 2010
a(n) = A022272(n) + A022272(-n). - Bruno Berselli, Mar 31 2015
a(n) = t(6*n) - 6*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(6*n) - 6*A000217(n). - Bruno Berselli, Aug 31 2017
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/90.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/180.
Product_{n>=1} (1 + 1/a(n)) = sqrt(15)*sinh(Pi/sqrt(15))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(15)*sin(Pi/sqrt(15))/Pi. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 15*x*(1 + x)/(1 - x)^3.
E.g.f.: 15*x*(1 + x)*exp(x).
a(n) = n*A008597(n) = A195046(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A064762 a(n) = 21*n^2.

Original entry on oeis.org

0, 21, 84, 189, 336, 525, 756, 1029, 1344, 1701, 2100, 2541, 3024, 3549, 4116, 4725, 5376, 6069, 6804, 7581, 8400, 9261, 10164, 11109, 12096, 13125, 14196, 15309, 16464, 17661, 18900, 20181, 21504, 22869, 24276, 25725, 27216, 28749
Offset: 0

Views

Author

Roberto E. Martinez II, Oct 18 2001

Keywords

Comments

Number of edges in a complete 7-partite graph of order 7n, K_n,n,n,n,n,n,n.

Links

Crossrefs

Similar sequences are listed in A244630.
Cf. A000217, A000290, A033428, A033581, A033582, A033583.
Cf. A008603, A195049.

Programs

Formula

a(n) = 42*n + a(n-1) - 21 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = 21*A000290(n) = 7*A033428(n) = 3*A033582(n). - Omar E. Pol, Jul 03 2014
a(n) = t(7*n) - 7*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(7*n) - 7*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 21*x*(1 + x)/(1-x)^3.
E.g.f.: 21*x*(1 + x)*exp(x).
a(n) = n*A008603(n) = A195049(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A064763 a(n) = 28*n^2.

Original entry on oeis.org

0, 28, 112, 252, 448, 700, 1008, 1372, 1792, 2268, 2800, 3388, 4032, 4732, 5488, 6300, 7168, 8092, 9072, 10108, 11200, 12348, 13552, 14812, 16128, 17500, 18928, 20412, 21952, 23548, 25200, 26908, 28672, 30492, 32368, 34300, 36288, 38332
Offset: 0

Views

Author

Roberto E. Martinez II, Oct 18 2001

Keywords

Comments

Number of edges in a complete 8-partite graph of order 8n, K_n,n,n,n,n,n,n,n.
Sequence found by reading the line from 0, in the direction 0, 28, ..., in the square spiral whose vertices are the generalized 16-gonal numbers. - Omar E. Pol, Jul 03 2014

Links

Crossrefs

Similar sequences are listed in A244630.
Cf. A000217, A000290, A001105, A016742, A033428, A033581, A033582, A033583.
Cf. A144555, A135628.

Programs

Formula

a(n) = 56*n + a(n-1) - 28 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
a(n) = 28*A000290(n) = 14*A001105(n) = 7*A016742(n) = 4*A033582(n) = 2*A144555(n). - Omar E. Pol, Jul 03 2014
From Vincenzo Librandi, Mar 30 2015: (Start)
G.f.: 28*x*(1+x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = t(8*n) - 8*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(8*n) - 8*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 28*x*(1 + x)*exp(x).
a(n) = n*A135628(n). (End)

A066545 Number of spanning trees in the line graph of the product of two complete graph, each of order n, L(K_n x K_n).

Original entry on oeis.org

4, 782757789696, 391497025772177207236260602767731880976449536, 79571717825565862744861159703491334416072984127575634790474236302905519522005340085288960000000000000000000000
Offset: 2

Views

Author

Roberto E. Martinez II, Jan 07 2002

Keywords

Comments

a(2) = 2^2, a(3) = 2^30 * 3^6, a(4) = 2^99 * 3^31, a(5) = 2^314 * 5^22. - Gerald McGarvey, Oct 20 2007

Examples

			NumberOfSpanningTrees(L(K_2 x K_2)) = 4.

Programs

  • Mathematica
    NumberOfSpanningTrees[LineGraph[GraphProduct[CompleteGraph[n], CompleteGraph[n]]]] (* First load package DiscreteMath`Combinatorica` *)

Extensions

Edited by Dean Hickerson, Jan 14 2002

A066544 Number of spanning trees in the line graph of the product of two star graphs, each of order n, L(S_n x S_n).

Original entry on oeis.org

4, 69360, 25181448044544, 500282053019322336000000000, 1364205042837678184255639132540659302400000000, 1119704625219101611411719462621416231171361585800882437615771859939328
Offset: 2

Views

Author

Roberto E. Martinez II, Jan 07 2002

Keywords

Examples

			NumberOfSpanningTrees(L(S_3 x S_3)) = 69360

Programs

  • Mathematica
    NumberOfSpanningTrees[LineGraph[GraphProduct[Star[n], Star[n]]]] (* First load package DiscreteMath`Combinatorica` *)

Extensions

Edited by Dean Hickerson, Jan 14 2002

A066546 Number of spanning trees in the line graph of the box product of two (2 x n) grid graphs.

Original entry on oeis.org

4, 5976745079881894723584, 22561207271999971793667241231967232556265564782280146786713600000000, 7757960313423917565831233144393761302830112940189177166113578796780294097560917685262512568386612793431012933632000000000000000000000000
Offset: 1

Views

Author

Roberto E. Martinez II, Jan 07 2002

Keywords

Links

Programs

  • Mathematica
    NumberOfSpanningTrees[LineGraph[GraphProduct[GridGraph[2, n], GridGraph[2, n]]]] (* First load package DiscreteMath`Combinatorica` *)

Extensions

Edited by Dean Hickerson, Jan 14 2002
More terms from Sean A. Irvine, Jul 08 2025

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