A380986 Product of prime indices of n (with multiplicity) minus product of distinct prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 12, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Feb 14 2025

Keywords

• nonn

Comments

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.

Examples

			The prime indices of 300 are {1,1,2,3,3}, so a(300) = 18 - 6 = 12.
		

Crossrefs

Positions of nonzeros are A038838.

For length instead of product we have A046660.

For factors instead of indices we have A066503, see A007947 (squarefree kernel).

For sum of factors instead of product of indices we have A280292, see A280286, A381075.

For quotient instead of difference we have A290106, for factors A003557.

For sum instead of product we have A380955 (firsts A380956, sorted A380957).

A000040 lists the primes, differences A001223.

A003963 gives product of prime indices, distinct A156061.

A005117 lists the squarefree numbers, complement A013929.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, length A001222.

A304038 lists distinct prime indices, sum A066328, length A001221.

Cf. A000720, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@prix[n]-Times@@Union[prix[n]],{n,100}]

Formula

a(n) = A003963(n) - A156061(n).