A376140 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.

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, 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, 1, 1, 1, 9, 1, 1, 4, 4, 1, 1, 1, 8, 4, 1, 1, 8, 1, 1, 1

Offset

1,4

Comments

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Formula

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

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)]).

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

a(n) < A000005(n).

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

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A325939 A318586 A222580 * A316978 A331023 A284345

Adjacent sequences: A376137 A376138 A376139 * A376141 A376142 A376143

Keyword

nonn,easy

Author

Amiram Eldar, Sep 11 2024

Status

approved