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A332775
a(n) = n + sopf(n) - omega(n).
1
1, 3, 5, 5, 9, 9, 13, 9, 11, 15, 21, 15, 25, 21, 21, 17, 33, 21, 37, 25, 29, 33, 45, 27, 29, 39, 29, 35, 57, 37, 61, 33, 45, 51, 45, 39, 73, 57, 53, 45, 81, 51, 85, 55, 51, 69, 93, 51, 55, 55, 69, 65, 105, 57, 69, 63, 77, 87, 117, 67, 121, 93, 71, 65, 81, 79, 133, 85, 93, 81, 141
OFFSET
1,2
COMMENTS
From Bernard Schott, Jun 10 2020: (Start)
All terms are odd, but not all odd integers are obtained: see A353046.
1 <= a(n) <= 2n-1 (see formula). (End)
FORMULA
a(n) = Sum_{k=1..n} k^(c(k)*(1 - ceiling(n/k) + floor(n/k)), where c is the prime characteristic (A010051).
a(n) = n + A055631(n).
From Bernard Schott, Jun 10 2020: (Start)
a(n) = 1 iff n = 1.
a(n) = 2*n-1 iff n is prime.
a(p^k) = p^k + p - 1 for p prime, k > 0. (End)
MATHEMATICA
Table[n - PrimeNu[n] + Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 100}]
PROG
(PARI) a(n) = n + vecsum(factor(n)[, 1]) - omega(n); \\ Michel Marcus, Jul 21 2020
CROSSREFS
Cf. A001221 (omega), A008472 (sopf), A010051, A055631, A353046.
Sequence in context: A212631 A090792 A076877 * A120841 A145282 A049757
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jun 08 2020
STATUS
approved