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Number of distinct parts that have neighbors in the integer partition with Heinz number n.
2

%I #13 Jan 28 2025 16:54:54

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

%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.

%C 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 Antti Karttunen, <a href="/A356735/b356735.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.

%F a(n) + A356733(n) = A001221(n).

%e The prime indices of 42 are {1,2,4}, of which 1 and 2 have neighbors, so a(42) = 2.

%e The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 4.

%e The prime indices of 990 are {1,2,2,3,5}, of which 1, 2, and 3 have neighbors, so a(990) = 3.

%e The prime indices of 1300 are {1,1,3,3,6}, none of which have neighbors, so a(1300) = 0.

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

%t Table[Length[Select[Union[primeMS[n]], MemberQ[primeMS[n],#-1]|| MemberQ[primeMS[n],#+1]&]],{n,100}]

%o (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 Positions of first appearances are A002110 without 1 (or A231209).

%Y The complement is counted by A356733.

%Y Positions of zeros are A356734.

%Y Positions of positive terms are A356736.

%Y A001221 counts distinct prime factors, sum A001414.

%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 A355393 counts partitions w/o a neighborless singleton, complement A356235.

%Y A355394 counts partitions w/o a neighborless part, complement A356236.

%Y A356226 lists the lengths of maximal gapless submultisets of prime indices:

%Y - length: A287170 (firsts A066205)

%Y - minimum: A356227

%Y - maximum: A356228

%Y - bisected length: A356229

%Y - standard composition: A356230

%Y - Heinz number: A356231

%Y - positions of first appearances: A356232

%Y Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356233, A356234, A356237.

%K nonn

%O 1,6

%A _Gus Wiseman_, Aug 31 2022

%E Data section extended to a(105) by _Antti Karttunen_, Jan 28 2025