A380958 Number of prime factors of n (with multiplicity) minus sum of distinct prime exponents of n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Feb 13 2025

Keywords

• nonn

Examples

			The prime factors of 2100 are {2,2,3,5,5,7}, with distinct multiplicities {1,2}, so a(2100) = 6 - (1+2) = 3.
		

Crossrefs

Positions of 0's are A130091, complement A130092.

The RHS (sum of distinct prime exponents) is A136565.

For prime factors instead of exponents see A280292, firsts A280286, sorted A381075.

For prime indices instead of exponents see A380955, firsts A380956, sorted A380957.

Position of first appearance of n is A380989(n).

A000040 lists the primes, differences A001223.

A005117 lists squarefree numbers, complement A013929.

A005361 gives product of prime signature.

A055396 gives least prime index, greatest A061395.

A056239 (reverse A296150) adds up prime indices, row sums of A112798, counted by A001222.

A124010 lists prime exponents (signature); see A001222, A001221, A051903, A051904.

Cf. A000720, A046660, A071625, A075254, A075255, A076694, A081770, A116861, A178503, A290106, A380986.

Programs

Mathematica

Table[PrimeOmega[n]-Total[Union[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

Formula

a(n) = A001222(n) - A136565(n).