A394651 - OEIS

%I #15 May 15 2026 18:02:27

%S 1,3,4,3,6,7,8,4,3,9,12,8,14,11,10,5,18,7,20,10,12,15,24,10,3,17,4,12,

%T 30,15,32,6,16,21,14,9,38,23,18,12,42,17,44,16,10,27,48,12,3,7,22,18,

%U 54,9,18,14,24,33,60,18,62,35,12,7,20,21,68,22,28,19,72,12,74,41,8,24,20,23,80,14,5,45,84,20,24

%N a(n) = tau(n) + Sum_{p|n, p prime, gcd(p,n/p) = 1} (p-1).

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

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

%F a(n) = A000005(n) + A063956(n) - A056169(n).

%F a(p^k) = k + p^floor(1/k) for p prime and k >= 1. - _Wesley Ivan Hurt_, May 15 2026

%e a(30) = 1^0 + 2^1 + 3^1 + 5^1 + 6^0 + 10^0 + 15^0 + 30^0 = 15.

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

%o (PARI) a(n) = my(f=factor(n)); numdiv(f) + sum(k=1, #f~, my(p=f[k,1]); if (gcd(p, n/p) == 1, p-1)); \\ _Michel Marcus_, Apr 25 2026

%Y Cf. A000005 (tau), A010051, A056169, A063956.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Apr 24 2026