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

Author

Gus Wiseman, Nov 02 2022

Keywords

• nonn

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.

Examples

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

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.

Programs

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}]

Formula

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