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

1,4

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.

Links

Table of n, a(n) for n=1..78.

Formula

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

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

Example

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

Mathematica

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

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.

Sequence in context: A128219 A238969 A238956 * A295511 A116505 A110534

Adjacent sequences: A331412 A331413 A331414 * A331416 A331417 A331418

Keyword

nonn

Author

Gus Wiseman, Jan 17 2020

Status

approved