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A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.

%I A350845 #9 Jan 27 2022 20:46:51
%S A350845 6,12,18,21,24,30,36,42,48,54,60,63,65,66,72,78,84,90,96,102,108,114,
%T A350845 120,126,130,132,133,138,144,147,150,156,162,168,174,180,186,189,192,
%U A350845 195,198,204,210,216,222,228,231,234,240,246,252,258,260,264,266,270
%N A350845 Heinz numbers of integer partitions with at least two adjacent parts of quotient 2.
%C A350845 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at least two adjacent prime indices of quotient 1/2.
%e A350845 The terms and corresponding partitions begin:
%e A350845    6: (2,1)
%e A350845   12: (2,1,1)
%e A350845   18: (2,2,1)
%e A350845   21: (4,2)
%e A350845   24: (2,1,1,1)
%e A350845   30: (3,2,1)
%e A350845   36: (2,2,1,1)
%e A350845   42: (4,2,1)
%e A350845   48: (2,1,1,1,1)
%e A350845   54: (2,2,2,1)
%e A350845   60: (3,2,1,1)
%e A350845   63: (4,2,2)
%e A350845   65: (6,3)
%e A350845   66: (5,2,1)
%e A350845   72: (2,2,1,1,1)
%e A350845   78: (6,2,1)
%e A350845   84: (4,2,1,1)
%e A350845   90: (3,2,2,1)
%e A350845   96: (2,1,1,1,1,1)
%t A350845 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A350845 Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]
%Y A350845 The complement is A350838, counted by A350837.
%Y A350845 The strict complement is counted by A350840.
%Y A350845 These partitions are counted by A350846.
%Y A350845 A000041 = integer partitions.
%Y A350845 A000045 = sets containing n with all differences > 2.
%Y A350845 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A350845 A116931 = partitions with no successions, ranked by A319630.
%Y A350845 A116932 = partitions with differences != 1 or 2, strict A025157.
%Y A350845 A323092 = double-free integer partitions, ranked by A320340.
%Y A350845 A325160 ranks strict partitions with no successions, counted by A003114.
%Y A350845 A350839 = partitions with gaps and conjugate gaps, ranked by A350841.
%Y A350845 Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.
%K A350845 nonn
%O A350845 1,1
%A A350845 _Gus Wiseman_, Jan 20 2022