A317676 Triangle whose n-th row lists in order all e-numbers of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

1, 2, 3, 4, 5, 6, 8, 9, 16, 7, 10, 12, 13, 21, 25, 27, 32, 36, 64, 81, 128, 256, 11, 14, 17, 18, 28, 33, 35, 41, 45, 49, 75, 93, 100, 125, 144, 145, 169, 216, 243, 279, 441, 512, 625, 729, 1024, 1296, 2048, 2187, 4096, 6561, 8192, 16384, 65536, 524288, 8388608, 9007199254740992

Offset

1,2

Comments

Given a positive integer n we construct a unique free pure symmetric multifunction e(n) by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)].

Every free pure symmetric multifunction (with empty expressions allowed) f with one atom and n positions has a unique e-number n such that e(n) = f, and vice versa, so this sequence is a permutation of the positive integers.

Links

Table of n, a(n) for n=1..58.

Mathematica Reference, Orderless

Example

Triangle begins:
  1
  2
  3   4
  5   6   8   9  16
  7  10  12  13  21  25  27  32  36  64  81 128 256
Corresponding triangle of free pure symmetric multifunctions (with empty expressions allowed) begins:
  o,
  o[],
  o[][], o[o],
  o[][][], o[o][], o[o[]], o[][o], o[o,o].

Mathematica

maxUsing[n_]:=If[n==1, {"o"}, Join@@Cases[Table[PR[k, n-k-1], {k, n-1}], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{maxUsing[h], Union[Sort/@Tuples[maxUsing/@p]]}], {p, IntegerPartitions[g]}]]];
radQ[n_]:=And[n>1, GCD@@FactorInteger[n][[All, 2]]==1];
Clear[rad]; rad[n_]:=rad[n]=If[n==0, 1, NestWhile[#+1&, rad[n-1]+1, Not[radQ[#]]&]];
ungo[x_?AtomQ]:=1; ungo[h_[g___]]:=rad[ungo[h]]^(Times@@Prime/@ungo/@{g});
Table[Sort[ungo/@maxUsing[n]], {n, 5}]

Crossrefs

Row lengths are A277996.

Cf. A007916, A052409, A052410, A052893, A053492, A215366, A279944, A280000, A299759, A317658, A317659, A317677.

Sequence in context: A259623 A305734 A002242 * A039125 A203616 A274049

Adjacent sequences: A317673 A317674 A317675 * A317677 A317678 A317679

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 03 2018

Status

approved