A356231 Heinz number of the sequence (A356226) of lengths of maximal gapless submultisets of the prime indices of n.

1, 2, 2, 3, 2, 3, 2, 5, 3, 4, 2, 5, 2, 4, 3, 7, 2, 5, 2, 6, 4, 4, 2, 7, 3, 4, 5, 6, 2, 5, 2, 11, 4, 4, 3, 7, 2, 4, 4, 10, 2, 6, 2, 6, 5, 4, 2, 11, 3, 6, 4, 6, 2, 7, 4, 10, 4, 4, 2, 7, 2, 4, 6, 13, 4, 6, 2, 6, 4, 6, 2, 11, 2, 4, 5, 6, 3, 6, 2, 14, 7, 4, 2, 10

Offset

1,2

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.

A multiset is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Links

Table of n, a(n) for n=1..84.

Formula

A001222(a(n)) = A287170(n).

A055396(a(n)) = A356227(n).

A061395(a(n)) = A356228(n).

Example

The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}. These have lengths (3,1,2), with Heinz number 30, so a(18564) = 30.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@Prime/@Length/@Split[primeMS[n], #1>=#2-1&], {n, 100}]

Crossrefs

Positions of prime terms are A073491, complement A073492.

Positions of terms with bigomega 2-4 are A073493-A073495.

Applying bigomega gives A287170, firsts A066205, even bisection A356229.

These are the Heinz numbers of the rows of A356226.

Minimal/maximal prime indices are A356227/A356228.

A version for standard compositions is A356230, firsts A356232/A356603.

A001221 counts distinct prime factors, with sum A001414.

A003963 multiplies together the prime indices.

A056239 adds up the prime indices, row sums of A112798.

A132747 counts non-isolated divisors, complement A132881.

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

Cf. A000005, A001222, A055932, A060680-A060683, A193829, A286470, A328166, A356233-A356237.

Sequence in context: A046216 A105560 A331597 * A339502 A342821 A165916

Adjacent sequences: A356228 A356229 A356230 * A356232 A356233 A356234

Keyword

nonn

Author

Gus Wiseman, Aug 18 2022

Status

approved