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 1-6 of 6 results.

A133459 Sums of two nonzero pentagonal pyramidal numbers.

Original entry on oeis.org

2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, 294, 306, 322, 328, 363, 392, 406, 411, 414, 423, 445, 480, 484, 531, 551, 556, 568, 576, 590, 601, 625, 676, 693, 727, 732, 744, 746, 766
Offset: 1

Views

Author

Jonathan Vos Post, Dec 23 2007

Keywords

Comments

Does this sequence ever include a pentagonal pyramidal number? That is, is it ever the case that A002411(i)=A002411(j)+A002411(k) as is often true for triangular pyramidal numbers (tetrahedral numbers) or square pyramidal numbers?
The answer to the above question is yes: A002411(30) + A002411(36) = 13950 + 23976 = 37926 = A002411(42) (see A172425). - Chai Wah Wu, Apr 16 2016

Crossrefs

Cf. A002311, A002411, A053721, A172425.

Programs

  • Mathematica
    nn = 12; Take[Union@ Map[Total, Tuples[#^2 (# + 1)/2 &@ Range@ nn, 2]], # (# - 1)/2 &[nn - 1]] (* Michael De Vlieger, Apr 16 2016 *)

Formula

{A002411(i) + A002411(j) for i, j > 0} = {i^2*(i+1)/2 + j^2*(j+1)/2 for i, j > 0}.

A053719 Let Py(n)=A000330(n)=n-th square pyramidal number. Consider all integer triples (i,j,k), j >= k>0, with Py(i)=Py(j)+Py(k), ordered by increasing i; sequence gives i values.

Original entry on oeis.org

55, 70, 147, 226, 237, 275, 351, 409, 434, 610, 714, 717, 869, 934, 1085, 1369, 1490, 1602, 1643, 1954, 2363, 2405, 2534, 3020, 3241, 3450, 4017, 4039, 4060, 4140, 4796, 5766, 5830, 6412, 8601, 8635, 8855, 8885, 9423, 10083, 10224, 10809, 11115, 11935
Offset: 0

Views

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 11 2000

Keywords

Comments

j values are A053720 and k values are A053721

Examples

			Py(55) = 56980 = Py(45) + Py(42); Py(70) = 116795 = Py(69) + Py(24);

Crossrefs

Cf. A000330, A053720, A053721.

Programs

  • Mathematica
    r[i_, j_] := Reduce[ j >= k > 0 && (2i + 1)*(i + 1)*i == (2j + 1)*(j + 1)*j + (2k + 1)*(k + 1)*k, k, Integers]; ijk = Reap[ Do[ If[ r[i, j] =!= False, sol = {i, j, k} /. ToRules[r[i, j]]; Print[sol]; Sow[sol]], {i, 1, 12000}, {j, Floor[4i/5], i-1}]][[2, 1]]; A053719 = ijk[[All, 1]]; A053720 = ijk[[All, 2]]; A053721 = ijk[[All, 3]]; (* Jean-François Alcover, Oct 17 2012 *)

Extensions

Crossrefs in comments corrected by Jean-François Alcover, Oct 17 2012

A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

Original entry on oeis.org

36, 81, 144, 289, 484, 576, 625, 676, 3600, 7396, 9801, 14400, 35344, 40000, 40804, 44100, 45796, 56644, 59049, 71824, 112896, 121104, 172225, 226576, 231361, 254016, 274576, 290521, 319225, 362404, 480249, 495616, 518400, 527076, 535824
Offset: 1

Views

Author

Alexander Adamchuk, Dec 25 2007

Keywords

Comments

Corresponding numbers m such that m^2 = a(n) are listed in A136360.
Note that some numbers in A136360 are also perfect squares. The corresponding numbers k such that m = k^2 are listed in A136361.
Includes all nonzero members of A099764: this occurs when the two pentagonal pyramidal numbers are both equal to i^2*(i+1)/2 where i+1 is a square. - Robert Israel, Feb 04 2020

Examples

			A133459 begins {2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, ...}.
Thus a(1) = 36, a(2) = 81, a(3) = 144, a(4) = 289 that are the perfect squares in A133459.

Links

Crossrefs

Cf. A136360, A136361, A133459, A002311, A002411, A053721, A099764.

Programs

  • Maple
    N:= 200: # for terms up to N^2*(N+1)/2.
PP:= [seq(i^2*(i+1)/2, i=1..N)]:
PP2:= sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],j=i..N),i=1..N)},PP[-1]),list)):
select(issqr,PP2); # Robert Israel, Feb 04 2020
  • Mathematica
    Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i,1,300}, {j,1,i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ]

Formula

a(n) = A136360(n)^2.

A053720 Let Py(n)=A000330(n)=n-th square pyramidal number. Consider all integer triples (i,j,k), j >= k>0, with Py(i)=Py(j)+Py(k), ordered by increasing i; sequence gives j values.

Original entry on oeis.org

45, 69, 145, 212, 225, 224, 344, 395, 377, 522, 643, 715, 845, 909, 1082, 1292, 1479, 1547, 1363, 1830, 2290, 2204, 2315, 3017, 3195, 2804, 3293, 4035, 3642, 3394, 4047, 5084, 5309, 5550, 8406, 8631, 8697, 8073, 8728, 9940, 9005, 10804, 10471, 11571
Offset: 0

Views

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 11 2000

Keywords

Comments

i values are A053719 and k values are A053721

Examples

			Py(55) = 56980 = Py(45) + Py(42); Py(70) = 116795 = Py(69) + Py(24);

Crossrefs

Cf. A000330, A053719, A053721.

Extensions

Crossrefs in comments corrected by Jean-François Alcover, Oct 17 2012

A136360 Square roots of the perfect squares in A133459.

Original entry on oeis.org

6, 9, 12, 17, 22, 24, 25, 26, 60, 86, 99, 120, 188, 200, 202, 210, 214, 238, 243, 268, 336, 348, 415, 476, 481, 504, 524, 539, 565, 602, 693, 704, 720, 726, 732, 846, 899, 961, 965, 990, 1026, 1202, 1218, 1221, 1224, 1320, 1551, 1602, 1687, 1716, 1724, 1734
Offset: 1

Views

Author

Alexander Adamchuk, Dec 25 2007

Keywords

Comments

Corresponding squares in A133459 are listed in A136359(n) = a(n)^2.
Note that some numbers in a(n) are also perfect squares: m = k^2 = {9, 25, 961, 17424, ...}. The corresponding numbers k such that a(n) = k^2 are listed in A136361.

Examples

			A133459 begins {2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, ...}.
Thus a(1) = sqrt(36) = 6, a(2) = sqrt(81) = 9, a(3) = sqrt(144) = 12, a(4) = sqrt(289) = 17 that are the square roots of the perfect squares in A133459.

Crossrefs

Cf. A136359, A136361, A133459, A002311, A002411, A053721.

Programs

  • Mathematica
    Sqrt[ Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i,1,300}, {j,1,i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ] ]

Formula

a(n) = sqrt(A136359(n)).

A136361 Square roots of the perfect squares in A136360; or numbers k such that k^4 is in A133459 = the sums of two nonzero pentagonal pyramidal numbers.

Original entry on oeis.org

3, 5, 31, 132, 1068, 9672, 50664, 145060
Offset: 1

Views

Author

Alexander Adamchuk, Dec 25 2007

Keywords

Comments

Corresponding perfect squares in A136360 are a(n)^2 = {9, 25, 961, 17424, ...}. They correspond to the perfect fourth powers in A133459 = Sums of two nonzero pentagonal pyramidal numbers. a(n)^4 are the terms of A133459: {81, 525, 923521, 303595776, ...}. Note that the first three terms are prime.
a(9) > (5*10^20)^(1/4). - Donovan Johnson, Jun 12 2011

Crossrefs

Cf. A136359, A136360, A133459, A002311, A002411, A053721.

Extensions

Name corrected and a(5)-a(6) from Donovan Johnson, Nov 20 2010
a(7)-a(8) from Donovan Johnson, Jun 12 2011
Showing 1-6 of 6 results.

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