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A300273
Sorted list of Heinz numbers of collapsible integer partitions.
48
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).
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]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 01 2018
STATUS
approved