A358105 Unreduced denominator of the n-th divisible pair, where pairs are ordered by Heinz number. Lesser prime index of A318990(n).

1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 5, 1, 2, 4, 1, 1, 1, 1, 2, 1, 6, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 4, 1, 2, 1, 1, 7, 1, 1, 2, 3, 1, 5, 2, 1, 1, 2, 1, 1, 8, 1, 3, 4, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1

Offset

1,3

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.

Links

Table of n, a(n) for n=1..87.

Formula

A358103(n) = A358104(n)/a(n).

Example

The 12th divisible pair is (2,6) so a(12) = 2.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Join@@Table[Cases[primeMS[n], {x_, y_}/; Divisible[y, x]:>x, {0}], {n, 1000}]

Crossrefs

The divisible pairs are ranked by A318990, proper A339005.

For all semiprimes we have A338912, greater A338913.

The quotient of the pair is A358103.

The reduced version for all semiprimes is A358193, numerator A358192.

A000040 lists the primes.

A001222 counts prime indices, distinct A001221.

A001358 lists the semiprimes, squarefree A006881.

A003963 multiplies together prime indices.

A056239 adds up prime indices.

A318991 ranks divisor-chains.

Cf. A000720, A027751, A128301, A215366, A289508, A289509, A296150, A300912, A318992, A358106.

Sequence in context: A346175 A165357 A346741 * A210961 A250007 A048996

Adjacent sequences: A358102 A358103 A358104 * A358106 A358107 A358108

Keyword

nonn

Author

Gus Wiseman, Nov 02 2022

Status

approved