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A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

%I A354235 #10 Sep 05 2022 22:38:23
%S A354235 5,10,13,15,20,23,25,26,30,35,37,39,40,45,46,47,50,52,55,60,61,65,69,
%T A354235 70,73,74,75,78,80,85,89,90,91,92,94,95,100,103,104,105,110,111,113,
%U A354235 115,117,120,122,125,130,135,137,138,140,141,143,145,146,148,150
%N A354235 Heinz numbers of integer partitions with at least one part divisible by 3.
%C A354235 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.
%e A354235 The terms together with their prime indices begin:
%e A354235     5: {3}
%e A354235    10: {1,3}
%e A354235    13: {6}
%e A354235    15: {2,3}
%e A354235    20: {1,1,3}
%e A354235    23: {9}
%e A354235    25: {3,3}
%e A354235    26: {1,6}
%e A354235    30: {1,2,3}
%e A354235    35: {3,4}
%e A354235    37: {12}
%e A354235    39: {2,6}
%e A354235    40: {1,1,1,3}
%e A354235    45: {2,2,3}
%e A354235    46: {1,9}
%e A354235    47: {15}
%e A354235    50: {1,3,3}
%e A354235    52: {1,1,6}
%e A354235    55: {3,5}
%e A354235    60: {1,1,2,3}
%t A354235 Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]
%Y A354235 For 4 instead of 3 we have A046101, counted by A295342.
%Y A354235 This sequence ranks the partitions counted by A295341, compositions A335464.
%Y A354235 For 2 instead of 3 we have A324929 (and A013929), counted by A047967.
%Y A354235 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A354235 A004250 counts partitions with some part > 2, compositions A008466.
%Y A354235 A004709 lists numbers divisible by no cube, counted by A000726.
%Y A354235 A036966 lists 3-full numbers, counted by A100405.
%Y A354235 A046099 lists non-cubefree numbers.
%Y A354235 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A354235 A124010 gives prime signature, sorted A118914.
%Y A354235 A354234 counts partitions of n with at least one part divisible by k.
%Y A354235 Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.
%K A354235 nonn
%O A354235 1,1
%A A354235 _Gus Wiseman_, May 23 2022