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A356733 Number of neighborless parts in the integer partition with Heinz number n.

%I A356733 #12 Jan 28 2025 16:55:06
%S A356733 0,1,1,1,1,0,1,1,1,2,1,0,1,2,0,1,1,0,1,2,2,2,1,0,1,2,1,2,1,0,1,1,2,2,
%T A356733 0,0,1,2,2,2,1,1,1,2,0,2,1,0,1,2,2,2,1,0,2,2,2,2,1,0,1,2,2,1,2,1,1,2,
%U A356733 2,1,1,0,1,2,0,2,0,1,1,2,1,2,1,1,2,2,2,2,1,0,2,2,2,2,2,0,1,2,2,2,1,1,1,2,0
%N A356733 Number of neighborless parts in the integer partition with Heinz number n.
%C A356733 A part x is neighborless if neither x - 1 nor x + 1 are parts.
%C A356733 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.
%H A356733 Antti Karttunen, <a href="/A356733/b356733.txt">Table of n, a(n) for n = 1..65537</a>
%H A356733 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A356733 a(n) = A001221(n) - A356735(n).
%e A356733 The prime indices of 42 are {1,2,4}, of which only 4 is neighborless, so a(42) = 1.
%e A356733 The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 0.
%e A356733 The prime indices of 1300 are {1,1,3,3,6}, with neighborless parts {1,3,6}, so a(1300) = 3.
%t A356733 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356733 Table[Length[Select[Union[primeMS[n]],!MemberQ[primeMS[n],#-1]&&!MemberQ[primeMS[n],#+1]&]],{n,100}]
%o A356733 (PARI) A356733(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ _Antti Karttunen_, Jan 28 2025
%Y A356733 Positions of first appearances are 1 followed by A066205.
%Y A356733 Dominated by A287170 (firsts also A066205).
%Y A356733 Positions of terms > 0 are A356734.
%Y A356733 The complement is counted by A356735.
%Y A356733 A001221 counts distinct prime factors, sum A001414.
%Y A356733 A003963 multiplies together prime indices.
%Y A356733 A007690 counts partitions with no singletons, complement A183558.
%Y A356733 A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356733 A073491 lists numbers with gapless prime indices, complement A073492.
%Y A356733 A132747 counts non-isolated divisors, complement A132881.
%Y A356733 A355393 counts partitions w/o a neighborless singleton, complement A356235.
%Y A356733 A355394 counts partitions w/o a neighborless part, complement A356236.
%Y A356733 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356733 A356607 counts strict partitions w/ a neighborless part, complement A356606.
%Y A356733 Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234, A356237.
%K A356733 nonn
%O A356733 1,10
%A A356733 _Gus Wiseman_, Aug 26 2022
%E A356733 Data section extended to a(105) by _Antti Karttunen_, Jan 28 2025