A333177 -id:A333177 - cp's OEIS frontend

A130030 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * prime(d).

2, 1, 1, 2, 3, 6, 5, 7, 9, 14, 11, 15, 17, 20, 21, 22, 27, 20, 31, 23, 33, 38, 39, 20, 45, 48, 43, 35, 53, 6, 67, 47, 65, 64, 63, 25, 85, 78, 73, 34, 99, 20, 107, 63, 45, 94, 119, 35, 113, 56, 99, 73, 137, 54, 103, 54, 117, 134, 161, -1, 163, 136, 73, 96, 113, 24, 199, 107, 159, 12, 213

Offset: 1

Gary W. Adamson, May 02 2007

sign

Old name: A129691 * A000040.

A130029 = A054523 * A000040.

			a(4) = 2 = dot product of row 4 of A129691: (-1, -1, 0, 1) and the first four primes: (2, 3, 5, 7) = (-2, -3, 0, 7) = 2.

Cf. A000010, A000040, A001221, A007444, A007947, A023900, A054523, A076479, A129691, A130029, A333177.

A129691 as an infinite lower triangular matrix * A000040, the primes.
From Ilya Gutkovskiy, Mar 22 2020: (Start)
a(n) = Sum_{d|n} A023900(n/d) * prime(d).
Sum_{k=1..n} a(gcd(n,k)) = prime(n). (End)

New name and more terms from Ilya Gutkovskiy, Mar 22 2020

A333178 a(n) = Sum_{d|n, gcd(d, n/d) = 1} prime(d).

Original entry on oeis.org

2, 5, 7, 9, 13, 23, 19, 21, 25, 45, 33, 51, 43, 65, 65, 55, 61, 89, 69, 91, 97, 115, 85, 115, 99, 147, 105, 133, 111, 223, 129, 133, 175, 203, 179, 183, 159, 235, 215, 205, 181, 337, 193, 233, 233, 287, 213, 283, 229, 331, 299, 289, 243, 359, 301, 301, 343, 385, 279, 461

Offset: 1

Ilya Gutkovskiy, Mar 10 2020

nonn

Cf. A000040, A001221, A007445, A076479, A333177.

a[n_] := Sum[If[GCD[n/d, d] == 1, Prime[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

a(n) = sumdiv(n, d, if (gcd(d, n/d) ==1, prime(d))); \\ Michel Marcus, Mar 10 2020

Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * a(d) = prime(n).

cp's OEIS Frontend

A130030 a(n) = Sum_{d|n} (-1)^omega(n/d) * phi(rad(n/d)) * prime(d).

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A333178 a(n) = Sum_{d|n, gcd(d, n/d) = 1} prime(d).

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