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A174538 The smallest k such that the number of steps in the n Collatz sequences starting at k+i, i=0..n-1, is always prime.

%I A174538 #10 Jul 09 2012 07:52:08
%S A174538 3,3,3,60,246,560,560,560,4722,4722,6032,6666,13956,13956,13956,13956,
%T A174538 13956,13956,13956,13956,13956,13956,13956,13956,81488,81488,81488,
%U A174538 83840,89535,89535,89535,282880,282984,282984,282984,282984
%N A174538 The smallest k such that the number of steps in the n Collatz sequences starting at k+i, i=0..n-1, is always prime.
%C A174538 Indices of A006577 that start a run of at least n primes.
%C A174538 We find long sequences of primes, for example there are 55 primes starting at index 282984, namely 83, 101, 127, 251,...
%H A174538 Donovan Johnson, <a href="/A174538/b174538.txt">Table of n, a(n) for n = 1..619</a> (terms <= 10^10)
%H A174538 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A174538 A006577(3)=7, A006577(4)=2 and A006577(5)=5 are all prime, so a(1)=a(2)=a(3) =3 mark the start of that run.
%p A174538 A174538 := proc(n)
%p A174538         local k,allp;
%p A174538         for k from 1 do
%p A174538                 allp := true;
%p A174538                 for i from 0 to n-1 do
%p A174538                         if not isprime(A006577(k+i)) then
%p A174538                                 allp := false;
%p A174538                                 break;
%p A174538                         end if;
%p A174538                 end do:
%p A174538                 if allp then
%p A174538                         return k;
%p A174538                 end if;
%p A174538         end do:
%p A174538 end proc: # _R. J. Mathar_, Jul 08 2012
%Y A174538 Cf. A008908, A066906, A070165, A033491.
%K A174538 nonn
%O A174538 1,1
%A A174538 _Michel Lagneau_, Mar 21 2010
%E A174538 Edited by _R. J. Mathar_, Jul 08 2012