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
Examples
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), ...
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
Programs
-
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]]
Extensions
Edited by N. J. A. Sloane, Nov 09 2016
Comments