A362707 a(n) = Sum_{d|n, phi(d)|sigma(d)} (n-d).

0, 1, 2, 5, 4, 12, 6, 13, 14, 17, 10, 36, 12, 25, 26, 29, 16, 60, 18, 37, 38, 41, 22, 96, 24, 49, 50, 67, 28, 123, 30, 61, 62, 65, 34, 156, 36, 73, 74, 77, 40, 184, 42, 85, 116, 89, 46, 216, 48, 97, 98, 101, 52, 204, 54, 151, 110, 113, 58, 351, 60, 121, 122, 125, 64, 252, 66, 133

Offset: 1

Author

Wesley Ivan Hurt, Apr 30 2023

Keywords

• nonn
• easy

Comments

Total distance from n to each balanced divisor of n (see example).

Examples

			a(12) = 36; 12 has 5 balanced divisors 1,2,3,6,12 and the sum of their distances to n is (12-1)+(12-2)+(12-3)+(12-6)+(12-12) = 36.
		

Crossrefs

Cf. A000005 (tau), A000010 (phi), A020492 (balanced numbers), A351112, A351113, A351114.

Programs

Mathematica

a[n_] := DivisorSum[n, (n - #) &, Divisible[DivisorSigma[1, #], EulerPhi[#]] &]; Array[a, 100]

PARI

a(n) = sumdiv(n, d, if (!(sigma(d) % eulerphi(d)), n-d)); \\ Michel Marcus, Apr 30 2023

Formula

a(n) = Sum_{d|n} (n-d) * A351114(d).

a(n) = n*A351112(n) - A351113(n).