A352064 Irregular triangle T(n,k) where row n lists the positions of n in A275314.

1, 2, 3, 4, 6, 8, 5, 9, 12, 16, 10, 18, 24, 32, 7, 15, 20, 27, 36, 48, 64, 14, 30, 40, 54, 72, 96, 128, 21, 25, 28, 45, 60, 80, 81, 108, 144, 192, 256, 42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512, 11, 35, 63, 75, 84, 100, 112, 135, 180, 240, 243, 320, 324, 432, 576, 768, 1024

Offset: 1

Author

Michael De Vlieger, Mar 02 2022

Keywords

• nonn
• tabf

Comments

A table by Leonhard Euler.

Let L(n-1) be a partition of (n-1) whose parts m are restricted to predecessors of primes. Row n lists the products (m+1) for all such partitions L(n-1).

Greatest term in row n is 2^(n-1).

Least term in row p prime is p.

Examples

			Triangle begins:
   1;
   2;
   3,  4;
   6,  8;
   5,  9, 12, 16;
  10, 18, 24, 32;
   7, 15, 20, 27,  36,  48,  64;
  14, 30, 40, 54,  72,  96, 128;
  21, 25, 28, 45,  60,  80,  81, 108, 144, 192, 256;
  42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512;
  ...
Illustration of relationship of terms of row n and partitions of (n-1) such that all parts m are restricted to prime predecessors. We show the partitions in parentheses, adding 1 to each part m below in brackets to take the product. The products are terms in row n in this sequence.
      1 = (1);
          [2]
row 2:     2;
.
      2 = (2),    (1+1);
          [3]     [2*2]
row 3:     3,       4;
.
      3 = (2+1),  (1+1+1);
          [3*2]   [2*2*2]
row 4:     6,       8;
.
      4 = (4),    (2+2),    (2+1+1),     (1+1+1+1);
          [5]     [3*3]     [3*2*2]      [2*2*2*2]
row 5:     5,       9,        12,           16;
.
      5 = (4+1),  (2+2+1),  (2+1+1+1),   (1+1+1+1+1);
          [5*2]   [3*3*2]   [3*2*2*2]    [2*2*2*2*2]
row 6:    10,      18,        24,           32;
etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10673 (rows n = 1..40, flattened)
• Leonhard Euler, Tentamen novae theoriae mvsicae ex certissimis harmoniae principiis dilvcide expositae, Petropoli, ex typographia Academiae scientiarvm (1739), 41.

Crossrefs

Cf. A001414, A006093, A275314, A280954.

Programs

Mathematica

With[{n = 12}, Take[#, n] &@ Values@ KeySort@ PositionIndex@ Array[Total[Flatten[ConstantArray[#1 - 1, #2] & @@@ FactorInteger[#]]] &, 2^n]] // TableForm (* syntactically simple, or, more efficiently *)
f[m_] := Block[{s = {Prime@ PrimePi[m + 1] - 1}}, KeySort@ Merge[#, Identity] &@ Join[{1 -> {}}, Reap[Do[If[# <= m, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 1, s = DeleteCases[s, 1]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]]] &@ Total[s], {i, Infinity}]][[-1, -1]] ]]; Map[Union[Times @@ # & /@ #] &, Values@ f[40] + 1] // Flatten

Formula

A280954(n) = length of row n.