A350845 - OEIS

%I #9 Jan 27 2022 20:46:51

%S 6,12,18,21,24,30,36,42,48,54,60,63,65,66,72,78,84,90,96,102,108,114,

%T 120,126,130,132,133,138,144,147,150,156,162,168,174,180,186,189,192,

%U 195,198,204,210,216,222,228,231,234,240,246,252,258,260,264,266,270

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

%C 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 The terms and corresponding partitions begin:

%e    6: (2,1)

%e   12: (2,1,1)

%e   18: (2,2,1)

%e   21: (4,2)

%e   24: (2,1,1,1)

%e   30: (3,2,1)

%e   36: (2,2,1,1)

%e   42: (4,2,1)

%e   48: (2,1,1,1,1)

%e   54: (2,2,2,1)

%e   60: (3,2,1,1)

%e   63: (4,2,2)

%e   65: (6,3)

%e   66: (5,2,1)

%e   72: (2,2,1,1,1)

%e   78: (6,2,1)

%e   84: (4,2,1,1)

%e   90: (3,2,2,1)

%e   96: (2,1,1,1,1,1)

%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t Select[Range[100],MemberQ[Divide@@@Partition[primeptn[#],2,1],2]&]

%Y The complement is A350838, counted by A350837.

%Y The strict complement is counted by A350840.

%Y These partitions are counted by A350846.

%Y A000041 = integer partitions.

%Y A000045 = sets containing n with all differences > 2.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y A116931 = partitions with no successions, ranked by A319630.

%Y A116932 = partitions with differences != 1 or 2, strict A025157.

%Y A323092 = double-free integer partitions, ranked by A320340.

%Y A325160 ranks strict partitions with no successions, counted by A003114.

%Y A350839 = partitions with gaps and conjugate gaps, ranked by A350841.

%Y Cf. A000929, A001105, A018819, A045690, A045691, A094537, A154402, A319613, A323093, A337135, A342094, A342095, A342098, A342191.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 20 2022