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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
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.
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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
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 *)
Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.
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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
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.
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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
Square roots of the perfect squares in A133459.
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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
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.
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2
3, 5, 31, 132, 1068, 9672, 50664, 145060
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