A376141 The maximum exponent in the prime factorization of the numbers k such that k and k+1 have the same maximum exponent in their prime factorization.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Sep 11 2024

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A051903, A369020.

Programs

Mathematica

emax[n_] := Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; With[{t = Table[emax[n], {n, 1, 500}]}, t[[Position[Differences[t], 0] // Flatten]]]

PARI

emax(n) = if(n == 1, 0, vecmax(factor(n)[, 2]));
lista(kmax) = {my(e1 = 0, e2); for(k = 2, kmax, e2 = emax(k); if(e1 == e2, print1(e1, ", ")); e1 = e2);}

Formula

a(n) = A051903(A369020(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (d(2) + Sum_{k>=2} k * (d(k) + d(k+1) - 2 * d2(k)))/d0 = 1.14396758638154735362..., where d(k) = Product_{p prime} (1 - 2/p^k), d2(k) = Product_{p prime} (1 - 1/p^k - 1/p^(k+1)), and d0 = 0.36939178586283962461... is the asymptotic density of A369020.