login
A331415
Sum of prime factors minus sum of prime indices of n.
13
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.
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.
Sequence in context: A128219 A238969 A238956 * A295511 A116505 A110534
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 17 2020
STATUS
approved