A300273 Sorted list of Heinz numbers of collapsible integer partitions.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169

Offset

1,1

Comments

A positive integer is in this sequence iff it can be reduced to a prime number by a sequence of collapses, where a collapse is a replacement of prime(n)^k with prime(n*k) in a number's prime factorization (k > 1).

Links

Table of n, a(n) for n=1..62.

Example

A sequence of collapses is 84 -> 63 -> 49 -> 19 corresponding to the sequence of partitions (4211) -> (422) -> (44) -> (8). Hence 84 is in the sequence.

Mathematica

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
repcaps[q_]:=Union[{q}, If[SquareFreeQ[q], {}, Union@@repcaps/@Union[Times[q/#, Prime[Plus@@primeMS[#]]]&/@Select[Rest[Divisors[q]], !PrimeQ[#]&&PrimePowerQ[#]&]]]];
Select[Range[200], MemberQ[repcaps[#], _?PrimeQ]&]

Crossrefs

Cf. A000041, A001222, A002577, A005117, A018818, A056239, A094457, A112798, A145515, A215366, A275870, A289078, A289079, A291441, A296150, A299202.

Sequence in context: A326836 A326841 A274222 * A353844 A353833 A219301

Adjacent sequences: A300270 A300271 A300272 * A300274 A300275 A300276

Keyword

nonn

Author

Gus Wiseman, Mar 01 2018

Status

approved