Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #5 Mar 03 2018 14:57:48
%S 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,
%T 48,49,53,59,61,63,64,67,71,73,79,81,83,84,89,97,101,103,107,108,109,
%U 112,113,121,125,127,128,131,137,139,144,149,151,157,163,167,169
%N Sorted list of Heinz numbers of collapsible integer partitions.
%C 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).
%e 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.
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t repcaps[q_]:=Union[{q},If[SquareFreeQ[q],{},Union@@repcaps/@Union[Times[q/#,Prime[Plus@@primeMS[#]]]&/@Select[Rest[Divisors[q]],!PrimeQ[#]&&PrimePowerQ[#]&]]]];
%t Select[Range[200],MemberQ[repcaps[#],_?PrimeQ]&]
%Y Cf. A000041, A001222, A002577, A005117, A018818, A056239, A094457, A112798, A145515, A215366, A275870, A289078, A289079, A291441, A296150, A299202.
%K nonn
%O 1,1
%A _Gus Wiseman_, Mar 01 2018