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

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, 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, 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

Offset

1,6

Comments

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.

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to prime indices in the factorization of n.

Formula

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

Example

The prime indices of 42 are {1,2,4}, of which 1 and 2 have neighbors, so a(42) = 2.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 4.
The prime indices of 990 are {1,2,2,3,5}, of which 1, 2, and 3 have neighbors, so a(990) = 3.
The prime indices of 1300 are {1,1,3,3,6}, none of which have neighbors, so a(1300) = 0.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Union[primeMS[n]], MemberQ[primeMS[n], #-1]|| MemberQ[primeMS[n], #+1]&]], {n, 100}]

Prog

(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

Crossrefs

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

The complement is counted by A356733.

Positions of zeros are A356734.

Positions of positive terms are A356736.

A001221 counts distinct prime factors, sum A001414.

A007690 counts partitions with no singletons, complement A183558.

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

A073491 lists numbers with gapless prime indices, complement A073492.

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

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

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

- length: A287170 (firsts A066205)

- minimum: A356227

- maximum: A356228

- bisected length: A356229

- standard composition: A356230

- Heinz number: A356231

- positions of first appearances: A356232

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

Sequence in context: A126825 A045833 A117896 * A275962 A132976 A143840

Adjacent sequences: A356732 A356733 A356734 * A356736 A356737 A356738

Keyword

nonn

Author

Gus Wiseman, Aug 31 2022

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

Status

approved