A376140 - OEIS

%I #15 Sep 13 2024 10:14:52

%S 0,1,1,2,1,1,1,3,2,1,1,4,1,1,1,4,1,4,1,4,1,1,1,6,2,1,3,4,1,1,1,5,1,1,

%T 1,4,1,1,1,6,1,1,1,4,4,1,1,8,2,4,1,4,1,6,1,6,1,1,1,8,1,1,4,6,1,1,1,4,

%U 1,1,1,9,1,1,4,4,1,1,1,8,4,1,1,8,1,1,1

%N The number of divisors of n whose prime factorization has maximum exponent that is smaller than the maximum exponent in the prime factorization of n.

%C The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0.

%H Amiram Eldar, <a href="/A376140/b376140.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F a(n) = Sum_{d|n} [m(d) = m(n)], where m(n) = A051903(n) and [] is the Iverson bracket.

%F If n = Product_{i} p_i^e_i (where p_i are distinct primes), then a(n) = Product_{i} (e_i + [e_i < Max_{i}(e_i)]).

%F a(n) <= 1 if and only if n is squarefree (A005117), and a(n) = 0 only for n = 1.

%F a(n) < A000005(n).

%F a(n) = A000005(A375932(n)) * A005361(A375931(n)) for n >= 2.

%F a(n) = A000005(n) * (A051903(n)/(A051903(n)+1))^A362611(n) for n >= 2.

%t a[n_] := Module[{e = FactorInteger[n][[;;,2]], m}, m = Max[e]; Times@@ ((Min[#, m-1] & /@ e) + 1)]; a[1] = 0; Array[a, 100]

%o (PARI) a(n) = if(n == 1, 0, my(e = factor(n)[,2], m = vecmax(e)); vecprod(apply(x -> 1 + min(x, m-1), e)));

%Y Cf. A000005, A005117, A051903, A005361, A362611, A375931, A375932.

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, Sep 11 2024