A356736 - OEIS

%I #10 Sep 01 2022 09:33:46

%S 1,6,12,15,18,24,30,35,36,45,48,54,60,72,75,77,90,96,105,108,120,135,

%T 143,144,150,162,175,180,192,210,216,221,225,240,245,270,288,300,315,

%U 323,324,360,375,384,385,405,420,432,437,450,462,480,486,525,539,540

%N Heinz numbers of integer partitions with no neighborless parts.

%C First differs from A066312 in having 1 and lacking 462.

%C First differs from A104210 in having 1 and lacking 42.

%C A part x is neighborless iff neither x - 1 nor x + 1 are parts.

%C 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 The terms together with their prime indices begin:

%e    1: {}

%e    6: {1,2}

%e   12: {1,1,2}

%e   15: {2,3}

%e   18: {1,2,2}

%e   24: {1,1,1,2}

%e   30: {1,2,3}

%e   35: {3,4}

%e   36: {1,1,2,2}

%e   45: {2,2,3}

%e   48: {1,1,1,1,2}

%e   54: {1,2,2,2}

%e   60: {1,1,2,3}

%e   72: {1,1,1,2,2}

%e   75: {2,3,3}

%e   77: {4,5}

%e   90: {1,2,2,3}

%e   96: {1,1,1,1,1,2}

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

%t Select[Range[100],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

%Y These partitions are counted by A355394.

%Y The singleton case is the complement of A356237.

%Y The singleton case is counted by A355393, complement A356235.

%Y The strict complement is A356606, counted by A356607.

%Y The complement is A356734, counted by A356236.

%Y A000041 counts integer partitions, strict A000009.

%Y A001221 counts distinct prime factors, sum A001414.

%Y A003963 multiplies together the prime indices of n.

%Y A007690 counts partitions with no singletons, complement A183558.

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

%Y A073491 lists numbers with gapless prime indices, complement A073492.

%Y Cf. A066312, A286470, A287170 (firsts A066205), A328171, A328187, A328221, A328335, A356231, A356234.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 31 2022