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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).
7
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.
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&]
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, May 03 2018
STATUS
approved