A363636
Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.
Original entry on oeis.org
0, 3, 7, 13, 17, 18, 21, 31, 38, 43, 47, 57, 68, 73, 91, 99, 111, 117, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 268, 273, 293, 302, 307, 313, 322, 327, 343, 381, 413, 421, 438, 443, 463, 487, 507, 515, 553, 557, 577, 593, 601, 651, 693, 697, 703, 707
Offset: 1
0 is a term because 0^2+1 = 1 equals the empty product.
3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
38 is a term because 38^2+1 = 1445 = 5*17*17 = (2^2+1)*(4^2+1)^2. (This is the first term that requires more than two factors.)
Sequences that list those terms (or their indices or some other key) of a given sequence that are products of smaller terms of the same sequence (in other words, the nonprimitive terms of the multiplicative closure of the sequence):
-
g[lst_, p_] :=
Module[{t, i, j},
Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
multPartition[n_] :=
Module[{i, j, p, e, lst = {{}}}, {p, e} =
Transpose[FactorInteger[n]];
Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
output = Join[{0}, Flatten[Position[Table[
test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
, {n, 707}], True]]]
(* David Trimas, Jul 23 2023 *)
A374370
Square array read by antidiagonals: the n-th row lists n-gonal numbers that are products of smaller n-gonal numbers.
Original entry on oeis.org
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
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
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
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.)
Showing 1-3 of 3 results.
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