A369020 Numbers k such that k and k+1 have the same maximal exponent in their prime factorization.

2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 49, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 80, 82, 85, 86, 93, 94, 98, 99, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Author

Amiram Eldar, Jan 12 2024

Keywords

• nonn
• easy

Comments

Differs from A358817 by having the terms 99, 165, 166, ..., which are not in A358817, and not having the terms 1, 440, 1331, 1575, ..., which are in A358817.

Numbers k such that A051903(k) = A051903(k+1).

If k is a term then k*(k+1) is a term of A362605.

The asymptotic density of this sequence is d(2) + Sum_{k>=2} (d(k) + d(k+1) - 2 * d2(k)) = 0.36939178586283962461..., where d(k) = Product_{p prime} (1 - 2/p^k) and d2(k) = Product_{p prime} (1 - 1/p^k - 1/p^(k+1)).

Links

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

Crossrefs

Cf. A051903, A358817, A362605, A369022.

Subsequences: A007674, A071318, A369021.

Programs

Mathematica

emax[n_] := emax[n] = Max[FactorInteger[n][[;; , 2]]]; emax[1] = 0; Select[Range[200], emax[#] == emax[# + 1] &]

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(k-1, ", ")); e1 = e2);}