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A332775
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a(n) = n + sopf(n) - omega(n).
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1
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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
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OFFSET
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1,2
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COMMENTS
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All terms are odd, but not all odd integers are obtained: see A353046.
1 <= a(n) <= 2n-1 (see formula). (End)
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LINKS
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FORMULA
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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) = 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)
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MATHEMATICA
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Table[n - PrimeNu[n] + Sum[p, {p, Select[Divisors[n], PrimeQ]}], {n, 100}]
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PROG
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(PARI) a(n) = n + vecsum(factor(n)[, 1]) - omega(n); \\ Michel Marcus, Jul 21 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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