A374370 -id:A374370 - cp's OEIS frontend

A068143 Triangular numbers which are products of triangular numbers larger than 1.

36, 45, 210, 300, 378, 630, 780, 990, 1485, 1540, 2850, 3240, 3570, 4095, 4851, 4950, 5460, 8778, 9180, 11781, 15400, 17955, 19110, 21528, 25200, 26565, 26796, 31878, 33930, 37128, 37950, 39060, 40755, 43956, 52650, 61425, 61776, 70125, 79800, 82215, 97020

Offset: 1

Amarnath Murthy, Feb 23 2002

nonn

			210 is a term as 210 = 21*10 both triangular numbers.
300 is also a term since 300 = 3*10*10.

Pontus von Brömssen, Table of n, a(n) for n = 1..3597 (all terms below 10^12).

Cf. A000217 (triangular numbers), A374374.
Except the first term, row n=3 of A374370 and row n=2 of A374498.
A188630 is a subsequence (only 2 factors allowed).

Corrected and extended by Jon E. Schoenfield, Jul 23 2006

a(38)-a(41) from Pontus von Brömssen, Jul 02 2024

A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

Original entry on oeis.org

1, 36, 1, 45, 560, 1, 210, 19600, 4900, 1, 300, 43680, 513590, 56448, 1, 378, 45760, 333833500, 127008, 4750, 1, 630, 893200, 711410700, 259200, 1926049000, 58372180608, 1

Offset: 2

Pontus von Brömssen, Jul 09 2024

If there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

The first term in each row is 1, because 1 is an n-gonal pyramidal number for every n and it equals the empty product.

			Array begins:
  n\k| 1     2          3            4
  ---+--------------------------------
  2  | 1    36         45          210
  3  | 1   560      19600        43680
  4  | 1  4900     513590    333833500
  5  | 1 56448     127008       259200
  6  | 1  4750 1926049000 655578709500

Cf. A080851, A374370, A374499 (second column).

Rows: A068143 (n=2 except the first term), A364151 (n=3), A374500 (n=4), A374501 (n=5), A374502 (n=6).

A374372 Pentagonal numbers that are products of smaller pentagonal numbers.

Original entry on oeis.org

1, 10045, 11310, 20475, 52360, 197472, 230300, 341055, 367290, 836640, 2437800, 2939300, 3262700, 4048352, 4268110, 4293450, 4619160, 4816000, 5969040, 6192520, 6913340, 6997320, 8531145, 10933650, 12397000, 16008300, 18573282, 18816875, 21430710, 24383520

Offset: 1

Pontus von Brömssen, Jul 07 2024

nonn

There are infinitely many terms where the corresponding product has two factors. This can be seen by solving the equation A000326(x)=A000326(y)*A000326(z) for a fixed z for which a solution exists, leading to a generalized Pell equation. For example, z = 5 leads to the solutions (x,y) = (82,14), (1649982,278898), (33266933642,5623138102), ..., corresponding to the terms A000326(82) = 10045, A000326(1649982) = 4083660075495, A000326(33266933642) = 1660033310895213609425, ... in the sequence.

			1 is a term because it is a pentagonal number and equals the empty product.
10045 is a term because it is a pentagonal number and equals the product of the pentagonal numbers 35 and 287.
20475 is a term because it is a pentagonal number and equals the product of the pentagonal numbers 5, 35, and 117. (This is the first term that requires more than two factors.)

Row n=5 of A374370.
A188663 is a subsequence (only 2 factors allowed).
Cf. A000326.

A374371 Least n-gonal number that can be written as a product of two or more smaller n-gonal numbers, or 0 if no such number exists.

Original entry on oeis.org

4, 36, 16, 10045, 2850, 6426, 1408, 265926, 69300, 79135, 9504, 195615, 145236, 126630, 42120, 81356859, 9410205, 165550, 1379840, 11340, 3009069, 8321351148, 316200, 47555937, 218338146, 9042726, 822528, 12300400, 300186051, 46955700, 766737400, 206898615

Offset: 2

Pontus von Brömssen, Jul 07 2024

nonn

			For 2 <= n <= 33, the n-gonal number a(n) can be written as a product of smaller n-gonal numbers in the following ways:
   n |          a(n)
  ---+---------------------------
   2 |          4 = 2*2
   3 |         36 = 6*6
   4 |         16 = 4*4
   5 |      10045 = 35*287
   6 |       2850 = 15*190
   7 |       6426 = 34*189
   8 |       1408 = 8*176
   9 |     265926 = 46*5781
  10 |      69300 = 10*6930
  11 |      79135 = 95*833
  12 |       9504 = 33*288
  13 |     195615 = 115*1701
  14 |     145236 = 14*10374
  15 |     126630 = 15*42*201
  16 |      42120 = 45*936
  17 |   81356859 = 549*148191
  18 |    9410205 = 343*27435
  19 |     165550 = 175*946
  20 |    1379840 = 20*68992
  21 |      11340 = 21*540
  22 |    3009069 = 427*7047
  23 | 8321351148 = 23*66*5481786
  24 |     316200 = 136*2325
  25 |   47555937 = 351*135487
  26 |  218338146 = 26*8397621
  27 |    9042726 = 154*58719
  28 |     822528 = 28*29376
  29 |   12300400 = 764*16100
  30 |  300186051 = 13051*23001
  31 |   46955700 = 3060*15345
  32 |  766737400 = 5720*134045
  33 |  206898615 = 12615*16401

Second column of A374370.

Cf. A057145.

A374373 Hexagonal numbers that are products of smaller hexagonal numbers.

Original entry on oeis.org

1, 2850, 61776, 79800, 103740, 145530, 437580, 719400, 810901, 828828, 1050525, 1185030, 1788886, 2031120, 2162160, 2821500, 4250070, 4959675, 9217071, 12298320, 12457536, 16356340, 22899528, 23334696, 24890040, 30728880, 32800950, 45158256, 48565440, 58487520

Offset: 1

Pontus von Brömssen, Jul 07 2024

nonn

There are infinitely many terms where the corresponding product has two factors. This can be proved by solving a certain generalized Pell equation, as in A374372.

			1 is a term because it is a hexagonal number and equals the empty product.
2850 is a term because it is a hexagonal number and equals the product of the hexagonal numbers 15 and 190.
103740 is a term because it is a hexagonal number and equals the product of the hexagonal numbers 6, 91, and 190. (This is the first term that requires more than two factors.)

Row n=6 of A374370.

Cf. A000384, A374372.

cp's OEIS Frontend

A068143 Triangular numbers which are products of triangular numbers larger than 1.

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A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

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A374372 Pentagonal numbers that are products of smaller pentagonal numbers.

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A374371 Least n-gonal number that can be written as a product of two or more smaller n-gonal numbers, or 0 if no such number exists.

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A374373 Hexagonal numbers that are products of smaller hexagonal numbers.

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