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Search: a053721 -id:a053721
Displaying 1-6 of 6 results found. page 1
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A133459 Sums of two nonzero pentagonal pyramidal numbers. +10
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
LINKS
Table of n, a(n) for n=1..55.
FORMULA
{A002411(i) + A002411(j) for i, j > 0} = {i^2*(i+1)/2 + j^2*(j+1)/2 for i, j > 0}.
MATHEMATICA
nn = 12; Take[Union@ Map[Total, Tuples[#^2 (# + 1)/2 &@ Range@ nn, 2]], # (# - 1)/2 &[nn - 1]] (* Michael De Vlieger, Apr 16 2016 *)
CROSSREFS
Cf. A002311, A002411, A053721, A172425.
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Dec 23 2007
STATUS
approved
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. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
j values are A053720 and k values are A053721
LINKS
Table of n, a(n) for n=0..43.
EXAMPLE
Py(55) = 56980 = Py(45) + Py(42); Py(70) = 116795 = Py(69) + Py(24);
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 *)
CROSSREFS
Cf. A000330, A053720, A053721.
KEYWORD
easy,nice,nonn
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 11 2000
EXTENSIONS
Crossrefs in comments corrected by Jean-François Alcover, Oct 17 2012
STATUS
approved
A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
LINKS
Robert Israel, Table of n, a(n) for n = 1..985
FORMULA
a(n) = A136360(n)^2.
EXAMPLE
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.
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[ # ] ] & ]
CROSSREFS
Cf. A136360, A136361, A133459, A002311, A002411, A053721, A099764.
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Dec 25 2007
STATUS
approved
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. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
i values are A053719 and k values are A053721
LINKS
Table of n, a(n) for n=0..43.
EXAMPLE
Py(55) = 56980 = Py(45) + Py(42); Py(70) = 116795 = Py(69) + Py(24);
CROSSREFS
Cf. A000330, A053719, A053721.
KEYWORD
easy,nice,nonn
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 11 2000
EXTENSIONS
Crossrefs in comments corrected by Jean-François Alcover, Oct 17 2012
STATUS
approved
A136360 Square roots of the perfect squares in A133459. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
LINKS
Table of n, a(n) for n=1..52.
FORMULA
a(n) = sqrt(A136359(n)).
EXAMPLE
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.
MATHEMATICA
Sqrt[ Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i, 1, 300}, {j, 1, i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ] ]
CROSSREFS
Cf. A136359, A136361, A133459, A002311, A002411, A053721.
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Dec 25 2007
STATUS
approved
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. +10
2
3, 5, 31, 132, 1068, 9672, 50664, 145060 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
LINKS
Table of n, a(n) for n=1..8.
CROSSREFS
Cf. A136359, A136360, A133459, A002311, A002411, A053721.
KEYWORD
more,nonn
AUTHOR
Alexander Adamchuk, Dec 25 2007
EXTENSIONS
Name corrected and a(5)-a(6) from Donovan Johnson, Nov 20 2010
a(7)-a(8) from Donovan Johnson, Jun 12 2011
STATUS
approved
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Last modified August 30 04:38 EDT 2024. Contains 375526 sequences. (Running on oeis4.)