A342028 - OEIS

%I #10 Feb 26 2021 06:30:41

%S 1,2,3,4,7,8,11,12,16,17,18,19,23,24,27,28,31,40,43,44,47,48,49,52,53,

%T 63,67,71,72,75,79,80,88,96,97,98,103,107,108,112,116,124,127,135,136,

%U 147,148,151,152,162,163,171,172,175,188,191,192,199,207,211,223

%N Numbers k such that k and k+1 both have mutually distinct exponents in their prime factorization (A130091).

%H Amiram Eldar, <a href="/A342028/b342028.txt">Table of n, a(n) for n = 1..10000</a>

%H Kevser Aktaş and M. Ram Murty, <a href="https://doi.org/10.1007/s12044-016-0326-z">On the number of special numbers</a>, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; <a href="https://www.ias.ac.in/article/fulltext/pmsc/127/03/0423-0430">alternative link</a>.

%H Bernardo Recamán Santos, <a href="http://mathoverflow.net/questions/201489">Consecutive numbers with mutually distinct exponents in their canonical prime factorization</a>, MathOverflow, Mar 30 2015.

%e 2 is a term since both 2 and 3 have a single exponent (1) in their prime factorization.

%e 5 is not a term since 6 = 2*3 has two equal exponents (1) in its prime factorization.

%t q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; Select[Range[250], q[#] && q[# + 1] &]

%Y Subsequence of A130091.

%Y Subsequences: A342029, A342030, A342031.

%K nonn

%O 1,2

%A _Amiram Eldar_, Feb 25 2021