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A300273
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Sorted list of Heinz numbers of collapsible integer partitions.
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48
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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]&]
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CROSSREFS
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Cf. A000041, A001222, A002577, A005117, A018818, A056239, A094457, A112798, A145515, A215366, A275870, A289078, A289079, A291441, A296150, A299202.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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