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

Author

Gus Wiseman, May 03 2018

Keywords

• nonn
• tabf

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.

Examples

			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}
		

Crossrefs

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

Programs

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&]