A322902 Numbers whose prime indices are all proper powers of the same number.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169

Offset

1,2

Comments

A prime index of n is a number m such that prime(m) divides n.

A proper power of n is a number n^k for some positive integer k.

Also the union of A322903 and A000079.

Links

Table of n, a(n) for n=1..61.

Example

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (5), (6), (1111), (7), (8), (42), (9), (33), (222).

Mathematica

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

Crossrefs

Cf. A001597, A018819, A023893, A023894, A052410, A056239, A072720, A072721, A302242, A302593, A322900, A322901, A322903.

Sequence in context: A324769 A328867 A326536 * A302040 A302036 A357864

Adjacent sequences: A322899 A322900 A322901 * A322903 A322904 A322905

Keyword

nonn

Author

Gus Wiseman, Dec 30 2018

Status

approved