cp's OEIS Frontend

A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.

%I A384723 #8 Jun 10 2025 16:25:58
%S A384723 1,2,4,6,12,18,24,30,60,90,120,150,180,210,240,420,540,630,840,1050,
%T A384723 1260,1470,1680,1890,2100,2310,2520,3360,4620,6300,6930,7560,9240
%N A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.
%C A384723 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A384723 Given a partition, the following are equivalent:
%C A384723 1) The distinct parts are maximally refined.
%C A384723 2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
%C A384723 3) No part is a sum of distinct non-parts.
%e A384723 The terms together with their prime indices begin:
%e A384723      1: {}
%e A384723      2: {1}
%e A384723      4: {1,1}
%e A384723      6: {1,2}
%e A384723     12: {1,1,2}
%e A384723     18: {1,2,2}
%e A384723     24: {1,1,1,2}
%e A384723     30: {1,2,3}
%e A384723     60: {1,1,2,3}
%e A384723     90: {1,2,2,3}
%e A384723    120: {1,1,1,2,3}
%e A384723    150: {1,2,3,3}
%e A384723    180: {1,1,2,2,3}
%e A384723    210: {1,2,3,4}
%e A384723    240: {1,1,1,1,2,3}
%e A384723    420: {1,1,2,3,4}
%e A384723    540: {1,1,2,2,2,3}
%e A384723    630: {1,2,2,3,4}
%e A384723    840: {1,1,1,2,3,4}
%t A384723 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384723 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A384723 nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
%t A384723 Select[Range[100],With[{y=conj[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]
%Y A384723 Partitions of this type are counted by A179009.
%Y A384723 The conjugate version is A383707, proper A384390.
%Y A384723 Appears to be the positions of 1 in A384005 (conjugate A383706).
%Y A384723 For at least one instead of exactly one choice we appear to have A384010.
%Y A384723 A055396 gives least prime index, greatest A061395.
%Y A384723 A056239 adds up prime indices, row sums of A112798.
%Y A384723 A122111 represents conjugation in terms of Heinz numbers.
%Y A384723 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384723 A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
%Y A384723 A357982 counts strict partitions of prime indices, non-strict A299200.
%Y A384723 Cf. A003963, A048767, A130091, A382525, A384317, A384318, A384320, A384347, A384349, A384394.
%K A384723 nonn,more
%O A384723 1,2
%A A384723 _Gus Wiseman_, Jun 09 2025