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A385378 The maximum possible number of distinct factors in the factorization of n into prime powers (A246655).

%I A385378 #10 Jun 29 2025 10:09:52
%S A385378 0,1,1,1,1,2,1,2,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,3,1,2,2,2,1,3,1,2,2,2,
%T A385378 2,2,1,2,2,3,1,3,1,2,2,2,1,3,1,2,2,2,1,3,2,3,2,2,1,3,1,2,2,3,2,3,1,2,
%U A385378 2,3,1,3,1,2,2,2,2,3,1,3,2,2,1,3,2,2,2
%N A385378 The maximum possible number of distinct factors in the factorization of n into prime powers (A246655).
%C A385378 Differs from A376885 and A384422 at n = 32, 64, 96, 128, 160, 192, ... .
%C A385378 Differs from A086435 at n = 36, 100, 144, 180, 196, 225, ... .
%C A385378 Differs from A375272 at n = 128, 384, 640, 896, 1024, 1152, ...	.
%C A385378 a(n) depends only on the prime signature of n (A118914).
%C A385378 The indices of records in this sequence are the partial products of the sequence of powers of primes (A000961), i.e., the terms in A024923.
%C A385378 The least index n such that a(n) = k, for k = 0, 1, 2, ..., is A024923(k+1).
%H A385378 Amiram Eldar, <a href="/A385378/b385378.txt">Table of n, a(n) for n = 1..10000</a>
%F A385378 Additive with a(p^e) = A003056(e).
%F A385378 a(n) >= A001221(n), with equality if and only if n is cubefree (A004709).
%F A385378 a(n) >= 1 for n >= 2, with equality if and only if n is a prime or a square of a prime (A000430).
%F A385378 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=2} P(k*(k+1)/2) = 0.19285739770001405035..., and P is the prime zeta function.
%e A385378       n | a(n) | factorization
%e A385378   ------+------+--------------------------------
%e A385378       2 |  1   | 2
%e A385378       6 |  2   | 2 * 3
%e A385378      24 |  3   | 2 * 3 * 2^2
%e A385378     120 |  4   | 2 * 3 * 2^2 * 5
%e A385378     840 |  5   | 2 * 3 * 2^2 * 5 * 7
%e A385378    6720 |  6   | 2 * 3 * 2^2 * 5 * 7 * 2^3
%e A385378   60480 |  7   | 2 * 3 * 2^2 * 5 * 7 * 2^3 * 3^2
%t A385378 f[p_, e_] := Floor[(Sqrt[8*e + 1] - 1)/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385378 (PARI) a(n) = vecsum(apply(x -> (sqrtint(8*x+1)-1)\2 , factor(n)[, 2]));
%Y A385378 Cf. A000430, A000961, A001221, A003056, A004709, A024923, A077761, A086435, A118914, A246655, A254578, A375272, A376885, A384422, A385379.
%K A385378 nonn,easy
%O A385378 1,6
%A A385378 _Amiram Eldar_, Jun 27 2025