A358103 Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).

1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 1, 4, 10, 11, 2, 12, 13, 14, 5, 15, 16, 6, 3, 17, 1, 18, 7, 2, 19, 20, 21, 22, 8, 23, 1, 24, 9, 4, 25, 26, 27, 10, 28, 29, 30, 5, 11, 31, 3, 32, 12, 33, 34, 1, 35, 36, 13, 6, 37, 2, 14, 38, 39, 15, 40, 41, 1, 42, 7, 4, 43

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) = 3.
		

Crossrefs

The divisible pairs are ranked by A318990, proper A339005.

Quotient of A358104 and A358105.

A different ordering is A358106.

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.

A358192/A358193 gives quotients of semiprime indices.

Cf. A000720, A027751, A032741, A215366, A289508, A289509, A296150, A300912, A318991, A318992.

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]:>y/x,{0}],{n,100}]

Formula

a(n) = A358104(n)/A358105(n).