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A356735 Number of distinct parts that have neighbors in the integer partition with Heinz number n.

%I A356735 #13 Jan 28 2025 16:54:54
%S A356735 0,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,
%T A356735 2,2,0,0,0,0,0,2,0,0,2,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,2,0,0,
%U A356735 0,2,0,2,0,0,2,0,2,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,2,0,0,3
%N A356735 Number of distinct parts that have neighbors in the integer partition with Heinz number n.
%C A356735 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.
%C A356735 Also the number of distinct prime indices x of n such that either x - 1 or x + 1 is also a prime index of n, where a prime index of n is a number x such that prime(x) divides n.
%H A356735 Antti Karttunen, <a href="/A356735/b356735.txt">Table of n, a(n) for n = 1..65537</a>
%H A356735 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A356735 a(n) + A356733(n) = A001221(n).
%e A356735 The prime indices of 42 are {1,2,4}, of which 1 and 2 have neighbors, so a(42) = 2.
%e A356735 The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 4.
%e A356735 The prime indices of 990 are {1,2,2,3,5}, of which 1, 2, and 3 have neighbors, so a(990) = 3.
%e A356735 The prime indices of 1300 are {1,1,3,3,6}, none of which have neighbors, so a(1300) = 0.
%t A356735 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356735 Table[Length[Select[Union[primeMS[n]], MemberQ[primeMS[n],#-1]|| MemberQ[primeMS[n],#+1]&]],{n,100}]
%o A356735 (PARI) A356735(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); omega(n)-sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ _Antti Karttunen_, Jan 28 2025
%Y A356735 Positions of first appearances are A002110 without 1 (or A231209).
%Y A356735 The complement is counted by A356733.
%Y A356735 Positions of zeros are A356734.
%Y A356735 Positions of positive terms are A356736.
%Y A356735 A001221 counts distinct prime factors, sum A001414.
%Y A356735 A007690 counts partitions with no singletons, complement A183558.
%Y A356735 A056239 adds up prime indices, row sums of A112798, lengths A001222.
%Y A356735 A073491 lists numbers with gapless prime indices, complement A073492.
%Y A356735 A355393 counts partitions w/o a neighborless singleton, complement A356235.
%Y A356735 A355394 counts partitions w/o a neighborless part, complement A356236.
%Y A356735 A356226 lists the lengths of maximal gapless submultisets of prime indices:
%Y A356735 - length: A287170 (firsts A066205)
%Y A356735 - minimum: A356227
%Y A356735 - maximum: A356228
%Y A356735 - bisected length: A356229
%Y A356735 - standard composition: A356230
%Y A356735 - Heinz number: A356231
%Y A356735 - positions of first appearances: A356232
%Y A356735 Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356233, A356234, A356237.
%K A356735 nonn
%O A356735 1,6
%A A356735 _Gus Wiseman_, Aug 31 2022
%E A356735 Data section extended to a(105) by _Antti Karttunen_, Jan 28 2025