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

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, 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, 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

Offset

1,10

Comments

A part x is neighborless if neither x - 1 nor x + 1 are parts.

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.

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) = A001221(n) - A356735(n).

Example

The prime indices of 42 are {1,2,4}, of which only 4 is neighborless, so a(42) = 1.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 0.
The prime indices of 1300 are {1,1,3,3,6}, with neighborless parts {1,3,6}, so a(1300) = 3.

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) 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

Crossrefs

Positions of first appearances are 1 followed by A066205.

Dominated by A287170 (firsts also A066205).

Positions of terms > 0 are A356734.

The complement is counted by A356735.

A001221 counts distinct prime factors, sum A001414.

A003963 multiplies together prime indices.

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.

A132747 counts non-isolated divisors, complement A132881.

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

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

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

A356607 counts strict partitions w/ a neighborless part, complement A356606.

Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234, A356237.

Sequence in context: A039999 A069842 A083056 * A321100 A244422 A061896

Adjacent sequences: A356730 A356731 A356732 * A356734 A356735 A356736

Keyword

nonn

Author

Gus Wiseman, Aug 26 2022

Extensions

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

Status

approved