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A342031 Starts of runs of 5 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091). 6

%I #11 Mar 08 2021 03:28:22

%S 1,16,241,2644,4372,1431124,12502348,112753348,750031648,2844282247,

%T 5882272324,6741230497,8004453748,87346072024,130489991521,

%U 218551872247,245127093748,460925878624,804065433748,1176638279524,2210511903748,2404792968748,2483167488748,3121595927521

%N Starts of runs of 5 consecutive numbers that have mutually distinct exponents in their prime factorization (A130091).

%C Bernardo Recamán Santos (2015) showed that there is no run of more than 23 consecutive numbers, since numbers of the form 36*k - 6 and 36*k + 6 do not have distinct exponents. Pace Nielsen and _Adam P. Goucher_ showed that there can be only finitely many runs of 23 consecutive numbers (see MathOverflow link).

%C Aktaş and Ram Murty (2017) gave an explicit upper bound to such a run of 23 numbers. They found the first 5 terms of this sequence (and stated that there are a few more known up to 7*10^8), and said that we may conjecture (based on numerical evidence) that there are no 6 consecutive numbers.

%H Martin Ehrenstein, <a href="/A342031/b342031.txt">Table of n, a(n) for n = 1..43</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 16 is a term since 16 = 2^4, 17, 18 = 2*3^2, 19 and 20 = 2^2*5 all have distinct exponents in their prime factorization.

%t q[n_] := Length[(e = FactorInteger[n][[;; , 2]])] == Length[Union[e]]; v = q /@ Range[5]; seq = {}; Do[If[And @@ v, AppendTo[seq, k - 5]]; v = Join[Rest[v], {q[k]}], {k, 6, 1.3*10^6}]; seq

%Y Subsequence of A130091, A342028, A342029 and A342030.

%K nonn

%O 1,2

%A _Amiram Eldar_, Feb 25 2021

%E a(15) and beyond from _Martin Ehrenstein_, Mar 08 2021

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)