A331415 Sum of prime factors minus sum of prime indices of n.

0, 1, 1, 2, 2, 2, 3, 3, 2, 3, 6, 3, 7, 4, 3, 4, 10, 3, 11, 4, 4, 7, 14, 4, 4, 8, 3, 5, 19, 4, 20, 5, 7, 11, 5, 4, 25, 12, 8, 5, 28, 5, 29, 8, 4, 15, 32, 5, 6, 5, 11, 9, 37, 4, 8, 6, 12, 20, 42, 5, 43, 21, 5, 6, 9, 8, 48, 12, 15, 6, 51, 5, 52, 26, 5, 13, 9, 9

Offset: 1

Author

Gus Wiseman, Jan 17 2020

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 factors of 12 are {2,2,3}, while the prime indices are {1,1,2}, so a(12) = 7 - 4 = 3.
		

Crossrefs

The number of k's is A331387(k) = sum of k-th column of A331385.

The sum of prime factors of n is A001414(n).

The sum of prime indices of n is A056239(n).

Numbers divisible by the sum of their prime factors are A036844.

Sum of prime factors is divisible by sum of prime indices: A331380

Product of prime indices equals sum of prime factors: A331384.

Cf. A000040, A000720, A001222, A014689, A056239, A112798, A301987, A318995, A325036, A330953, A331378, A331379, A331416, A331418.

Programs

Mathematica

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k*(p-PrimePi[p])]],{n,30}]

Formula

Totally additive with a(prime(k)) = prime(k) - k = A014689(k).

a(n) = A001414(n) - A056239(n).