A342030 - OEIS

%I #12 Feb 26 2021 06:30:54

%S 1,2,16,17,47,96,241,242,575,1249,2644,2645,4049,4372,4373,4799,9124,

%T 12248,33749,72250,120049,130436,281249,303748,1431124,1431125,

%U 1531250,2101247,3693761,4085656,4910975,12502348,12502349,14268481,22997761,25486324,26693549

%N Starts of runs of 4 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

%H Amiram Eldar, <a href="/A342030/b342030.txt">Table of n, a(n) for n = 1..196</a> (terms below 10^11)

%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 2, 3, 4 = 2^2, and 5 all have a single exponent in their prime factorization.

%e 3 is not a term since in the run {3, 4, 5, 6} the fourth member 6 = 2*3 has two equal exponents (1) in its prime factorization.

%t q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; v = q /@ Range[4]; seq = {}; Do[If[And @@ v, AppendTo[seq, k - 4]]; v = Join[Rest[v], {q[k]}], {k, 5, 10^5}]; seq

%Y Subsequence of A130091, A342028 and A342029.

%Y A342031 is a subsequence.

%K nonn

%O 1,2

%A _Amiram Eldar_, Feb 25 2021