A356734 - OEIS

%I #7 Aug 30 2022 09:41:50

%S 2,3,4,5,7,8,9,10,11,13,14,16,17,19,20,21,22,23,25,26,27,28,29,31,32,

%T 33,34,37,38,39,40,41,42,43,44,46,47,49,50,51,52,53,55,56,57,58,59,61,

%U 62,63,64,65,66,67,68,69,70,71,73,74,76,78,79,80,81,82,83

%N Heinz numbers of integer partitions with at least one neighborless part.

%C First differs from A319630 in lacking 1 and having 42 (prime indices: {1,2,4}).

%C A part x is neighborless if 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     2: {1}

%e     3: {2}

%e     4: {1,1}

%e     5: {3}

%e     7: {4}

%e     8: {1,1,1}

%e     9: {2,2}

%e    10: {1,3}

%e    11: {5}

%e    13: {6}

%e    14: {1,4}

%e    16: {1,1,1,1}

%e    17: {7}

%e    19: {8}

%e    20: {1,1,3}

%e    21: {2,4}

%e    22: {1,5}

%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 A356236.

%Y The singleton case is A356237, counted by A356235 (complement A355393).

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

%Y The complement is A356736, counted by A355394.

%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 A132747 counts non-isolated divisors, complement A132881.

%Y A356069 counts gapless divisors, initial A356224 (complement A356225).

%Y Cf. A000005, A286470, A287170 (firsts A066205), A289508, A325160, A328166, A328335, A356231, A356233, A356234.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 26 2022