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A385379 The maximum possible number of distinct composite prime powers (A246547) in the factorization of n into prime powers.

%I A385379 #7 Jun 29 2025 10:09:48
%S A385379 0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,1,0,1,1,0,0,0,2,0,0,
%T A385379 0,2,0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,1,2,0,0,0,1,
%U A385379 0,0,0,2,0,0,1,1,0,0,0,1,1,0,0,1,0,0,0
%N A385379 The maximum possible number of distinct composite prime powers (A246547) in the factorization of n into prime powers.
%C A385379 Differs from A376679 at n = 1, 48, 72, 80, ... .
%C A385379 The factorization includes primes if n is not a powerful number (A001694) that is larger than 1.
%C A385379 a(n) depends only on the prime signature of n (A118914).
%H A385379 Amiram Eldar, <a href="/A385379/b385379.txt">Table of n, a(n) for n = 1..10000</a>
%F A385379 Additive with a(p^e) = A052146(e+1).
%F A385379 a(n) = 0 if and only if n is squarefree (A005117).
%F A385379 a(A385380(n)) = n-1.
%F A385379 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=1} P(k*(k+3)/2) = 0.49006911093767425812..., and P is the prime zeta function.
%e A385379          n | a(n) | factorization
%e A385379   ---------+------+----------------------------------------
%e A385379          4 |  1   | 2^2
%e A385379         32 |  2   | 2^2 * 2^3
%e A385379        288 |  3   | 2^2 * 2^3 * 3^2
%e A385379       4608 |  4   | 2^2 * 2^3 * 3^2 * 2^4
%e A385379     115200 |  5   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2
%e A385379    3110400 |  6   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3
%e A385379   99532800 |  7   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3 * 2^5
%t A385379 f[p_, e_] := Floor[(Sqrt[8*e + 9] - 3)/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385379 (PARI) a(n) = vecsum(apply(x -> (sqrtint(8*x+9)-1)\2 , factor(n)[, 2]));
%Y A385379 Cf. A001694, A005117, A052146, A077761, A118914, A246547, A246655, A376679, A385378, A385380 (indices of records).
%K A385379 nonn,easy
%O A385379 1,32
%A A385379 _Amiram Eldar_, Jun 27 2025