A303945 Triangle whose n-th row lists the multiset of prime indices of the n-th number that is not a perfect power A007916(n).

1, 2, 3, 1, 2, 4, 1, 3, 5, 1, 1, 2, 6, 1, 4, 2, 3, 7, 1, 2, 2, 8, 1, 1, 3, 2, 4, 1, 5, 9, 1, 1, 1, 2, 1, 6, 1, 1, 4, 10, 1, 2, 3, 11, 2, 5, 1, 7, 3, 4, 12, 1, 8, 2, 6, 1, 1, 1, 3, 13, 1, 2, 4, 14, 1, 1, 5, 2, 2, 3, 1, 9, 15, 1, 1, 1, 1, 2, 1, 3, 3, 2, 7, 1, 1

Offset

1,2

Comments

A number is not a perfect power if its prime multiplicities are relatively prime. This triangle is an enumeration of all finite aperiodic multisets of positive integers, where a multiset is aperiodic if its multiplicities are relatively prime.

Links

Table of n, a(n) for n=1..85.

Example

Sequence of numbers that are not perfect powers together with their multisets of prime indices begins:
2:  {1}
3:  {2}
5:  {3}
6:  {1,2}
7:  {4}
10: {1,3}
11: {5}
12: {1,1,2}
13: {6}
14: {1,4}
15: {2,3}
17: {7}
18: {1,2,2}
19: {8}
20: {1,1,3}
21: {2,4}
22: {1,5}
23: {9}
24: {1,1,1,2}
26: {1,6}
28: {1,1,4}
29: {10}
30: {1,2,3}

Mathematica

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
primeMS/@Select[Range[2, 100], GCD@@FactorInteger[#][[All, 2]]===1&]

Crossrefs

Cf. A000041, A000720, A000837, A001222, A005117, A007916, A052409, A052410, A056239, A112798, A296150.

Sequence in context: A100833 A097293 A296656 * A026793 A344089 A329631

Adjacent sequences: A303942 A303943 A303944 * A303946 A303947 A303948

Keyword

nonn,tabf

Author

Gus Wiseman, May 03 2018

Status

approved