A395491 - OEIS

%I #5 May 11 2026 17:33:10

%S 1,3,4,3,6,14,8,4,3,22,12,17,14,30,32,5,18,23,20,25,44,46,24,31,3,54,

%T 4,33,30,95,32,6,68,70,72,9,38,78,80,47,42,131,44,49,50,94,48,57,3,55,

%U 104,57,54,61,112,63,116,118,60,130,62,126,68,7,132,203,68,73,140,215,72,12,74,150,80,81,156,239,80,89,5,166

%N a(n) = tau(n) + (n-1) * Sum_{p|n, p prime} [gcd(p,n/p) = 1], where [ ] is the Iverson bracket.

%C For each divisor d of n, add n if d is a unitary prime divisor of n, else add 1.

%F a(n) = Sum_{d|n} n^([gcd(d,n/d) = 1] * c(d)), where [ ] is the Iverson bracket and c = A010051.

%F a(n) = A000005(n) + (n-1) * A056169(n).

%F a(p^k) = k+1+floor(1/k)*(p-1) for p prime and k>=1. - _Wesley Ivan Hurt_, May 11 2026

%e a(24) = 24^0 + 24^0 + 24^1 + 24^0 + 24^0 + 24^0 + 24^0 + 24^0 = 31.

%t Table[Sum[n^(KroneckerDelta[GCD[d, n/d], 1]*(PrimePi[d] - PrimePi[d - 1])), {d, Divisors[n]}], {n, 100}]

%Y Cf. A000005 (tau), A010051, A056169, A394651.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Apr 25 2026