A054725 a(1)=1; a(n) = Sum_{p | n} e * a(p-1), where sum is over all primes p that divide n, and e is the multiplicity of p in n.

1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 4, 6, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 4, 6, 6, 5, 6, 5, 5, 6, 6, 5, 5, 6, 5, 6, 5, 6, 6, 5, 5, 6, 6, 6, 6, 6

Offset: 1

Author

Leroy Quet, Apr 20 2000

Keywords

• nonn

Examples

			a(20) = a(2-1) + a(2-1) + a(5-1) = 1 + 1 +2 = 4 because 20 = 2*2*5.
		

Links

• Joerg Arndt, Table of n, a(n) for n = 1..10000

Programs

Mathematica

Fold[Append[#1, Total@ Table[#1[[p - 1]], {p, Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#2]]}]] &, {1}, Range[2, 105]] (* Michael De Vlieger, Dec 11 2017 *)

PARI

a(n)=if (n<=1, 1, my(F=factor(n)); sum(e=1, #F[,1], F[e,2] * a(F[e,1]-1) ) );
vector(200, n, a(n)) \\ Joerg Arndt, Apr 08 2014

Formula

a(1) = 1 and a(n) = A064415(n) for n>=2. [Joerg Arndt, Apr 08 2014]

Extensions

Title clarified by Sean A. Irvine, Feb 18 2022