A374371 -id:A374371 - cp's OEIS frontend

A374370 Square array read by antidiagonals: the n-th row lists n-gonal numbers that are products of smaller n-gonal numbers.

1, 4, 1, 6, 36, 1, 8, 45, 16, 1, 9, 210, 36, 10045, 1, 10, 300, 64, 11310, 2850, 1, 12, 378, 81, 20475, 61776, 6426, 1, 14, 630, 100, 52360, 79800, 9828, 1408, 1, 15, 780, 144, 197472, 103740, 35224, 61920, 265926, 1, 16, 990, 196, 230300, 145530, 60606, 67200, 391950, 69300, 1

Offset: 2

Pontus von Brömssen, Jul 07 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 number for every n and it equals the empty product.

			Array begins:
   n=2: 1,      4,       6,       8,       9,       10,       12,       14
   n=3: 1,     36,      45,     210,     300,      378,      630,      780
   n=4: 1,     16,      36,      64,      81,      100,      144,      196
   n=5: 1,  10045,   11310,   20475,   52360,   197472,   230300,   341055
   n=6: 1,   2850,   61776,   79800,  103740,   145530,   437580,   719400
   n=7: 1,   6426,    9828,   35224,   60606,  1349460,  2077992,  3333330
   n=8: 1,   1408,   61920,   67200,  276640,   297045,   870485,  1022000
   n=9: 1, 265926,  391950, 1096200, 1767546,  1787500,  9909504, 28123200
  n=10: 1,  69300, 1297890, 4257000, 5756400,  9140040,  9729720, 10648800
  n=11: 1,  79135,  792330, 2382380, 5570565, 15361500, 22230000, 49888395
  n=12: 1,   9504,   45696,  604128, 1981980,  2208465,  4798080, 13837824

Cf. A057145, A374371 (second column), A374498.

Rows: A018252 (n=2), A068143 (n=3 except first term), A062312 (n=4), A374372 (n=5), A374373 (n=6).

A374499 Least n-gonal pyramidal number that can be written as a product of two or more smaller n-gonal pyramidal numbers, or 0 if no such number exists.

Original entry on oeis.org

36, 560, 4900, 56448, 4750, 58372180608, 1130220, 6252757280000

Offset: 2

Pontus von Brömssen, Jul 09 2024

a(11) = 4200, a(13) = 5521090680, a(14) = 748980.

			For 2 <= n <= 9, the n-gonal pyramidal number a(n) can be written as a product of smaller n-gonal pyramidal numbers in the following ways:
  n |             a(n)
  --+-------------------------------------
  2 |            36 = 6*6
  3 |           560 = 4*4*35 = 10*56
  4 |          4900 = 5*5*14*14
  5 |         56448 = 196*288
  6 |          4750 = 50*95
  7 |   58372180608 = 196*456*653108
  8 |       1130220 = 9*70*1794
  9 | 6252757280000 = 10*10*10*80*78159466

Second column of A374498.

Cf. A080851, A374371.

cp's OEIS Frontend

A374370 Square array read by antidiagonals: the n-th row lists n-gonal numbers that are products of smaller n-gonal numbers.

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