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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.
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%I #20 Feb 18 2022 20:44:39

%S 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,

%T 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,

%U 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

%N 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.

%H Joerg Arndt, <a href="/A054725/b054725.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 1 and a(n) = A064415(n) for n>=2. [_Joerg Arndt_, Apr 08 2014]

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

%t 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 *)

%o (PARI) a(n)=if (n<=1, 1, my(F=factor(n)); sum(e=1, #F[,1], F[e,2] * a(F[e,1]-1) ) );

%o vector(200, n, a(n)) \\ _Joerg Arndt_, Apr 08 2014

%K nonn

%O 1,4

%A _Leroy Quet_, Apr 20 2000

%E Title clarified by _Sean A. Irvine_, Feb 18 2022