A277564 Let {c(i)} = A007916 denote the sequence of numbers > 1 which are not perfect powers. Every positive integer n has a unique representation as a tower n = c(x_1)^c(x_2)^c(x_3)^...^c(x_k), where the exponents are nested from the right. The sequence is an irregular triangle read by rows, where the n-th row lists n followed by x_1, ..., x_k.

1, 2, 1, 3, 2, 4, 1, 1, 5, 3, 6, 4, 7, 5, 8, 1, 2, 9, 2, 1, 10, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 11, 16, 1, 1, 1, 17, 12, 18, 13, 19, 14, 20, 15, 21, 16, 22, 17, 23, 18, 24, 19, 25, 3, 1, 26, 20, 27, 2, 2, 28, 21, 29, 22, 30, 23, 31, 24, 32, 1, 3, 33, 25, 34, 26, 35, 27, 36, 4, 1, 37, 28, 38, 29, 39, 30, 40, 31

Offset

1,2

Comments

The row lengths are A288636(n) + 1. - Gus Wiseman, Jun 12 2017

See A278028 for a version in which row n simply lists x_1, x_2, ..., x_k (omitting the initial n).

Links

Gus Wiseman, Table of n, a(n) for n = 1..20131

N. J. A. Sloane, Maple programs for A007916, A278028, A278029, A052409, A089723, A277564

Example

1 is represented by the empty sequence (), by convention.
Successive rows of the triangle are as follows (c(k) denotes the k-th non-prime-power, A007916(k)):
2, 1,
3, 2,
4, 1, 1,
5, 3,
6, 4, because 6 = c(4)
7, 5,
8, 1, 2, because 8 = 2^3 = c(1)^c(2)
9, 2, 1,
10, 6,
11, 7,
...
16, 1, 1, 1, because 16 = 2^4 = c(1)^4 = c(1)^(c(1)^2) = c[1]^(c[1]^c[1])
17, 12,
...
This sequence represents a bijection N -> Q where Q is the set of all finite sequences of positive integers: 1->(), 2->(1), 3->(2), 4->(1 1), 5->(3), 6->(4), 7->(5), 8->(1 2), 9->(2 1), ...

Maple

See link.

Mathematica

nn=10000; radicalQ[1]:=False; radicalQ[n_]:=SameQ[GCD@@FactorInteger[n][[All, 2]], 1];
hyperfactor[1]:={}; hyperfactor[n_?radicalQ]:={n}; hyperfactor[n_]:=With[{g=GCD@@FactorInteger[n][[All, 2]]}, Prepend[hyperfactor[g], Product[Apply[Power[#1, #2/g]&, r], {r, FactorInteger[n]}]]];
rad[0]:=1; rad[n_?Positive]:=rad[n]=NestWhile[#+1&, rad[n-1]+1, Not[radicalQ[#]]&]; Set@@@Array[radPi[rad[#]]==#&, nn];
Flatten[Join[{#}, radPi/@hyperfactor[#]]&/@Range[nn]]

Crossrefs

Cf. A007916, A277562, A164336, A000961, A164337, A278028, A089723.

Sequence in context: A135550 A035491 A318574 * A363489 A363944 A367581

Adjacent sequences: A277561 A277562 A277563 * A277565 A277566 A277567

Keyword

nonn,tabf

Author

Gus Wiseman, Oct 20 2016

Extensions

Edited by N. J. A. Sloane, Nov 09 2016

Status

approved