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A276136 Numbers m > 1 such that the largest odd divisors of m-1, m, and m+1 are prime.

%I A276136 #29 Jun 27 2023 14:58:47
%S A276136 6,11,12,13,23,47,192,193,383,786432
%N A276136 Numbers m > 1 such that the largest odd divisors of m-1, m, and m+1 are prime.
%C A276136 Conjecture: this sequence is finite.
%C A276136 Any further terms are greater than 10^11. - _Charles R Greathouse IV_, Aug 22 2016
%C A276136 From _Robert Israel_, Apr 27 2020: (Start)
%C A276136 Each term is either of the form 3*2^k with 3*2^k-1 and 3*2^k+1 prime, or 3*2^k-1 with 3*2^k-1 prime and 3*2^(k-1)-1 prime, or 3*2^k+1 with 3*2^k+1 prime and 3*2^(k-1)+1 prime.
%C A276136 Any further terms > 10^2000.
%C A276136 (End)
%F A276136 A038550(a(n-1)) + 1 = A038550(a(n)) = A038550(a(n+1)) - 1.
%F A276136 a(n) >> n log n. - _Charles R Greathouse IV_, Aug 22 2016
%e A276136 6 is in this sequence because the largest odd divisor of 5 is 5, the largest odd divisor of 6 is 3 and the largest odd divisor of 7 is 7, and all three are prime.
%p A276136 Res:= 6:
%p A276136 for k from 2  while length(3*2^k-1)<1000 do
%p A276136   if (isprime(3*2^k-1) and isprime(3*2^(k-1)-1)) then Res:= Res, 3*2^k-1
%p A276136     fi;
%p A276136   if (isprime(3*2^k-1) and isprime(3*2^k+1)) then Res:= Res, 3*2^k;
%p A276136     fi;
%p A276136   if (isprime(3*2^k+1) and isprime(3*2^(k-1)+1)) then Res:= Res, 3*2^k+1;
%p A276136     fi;
%p A276136 od:
%p A276136 Res; # _Robert Israel_, Apr 27 2020
%t A276136 Select[Range[2, 10^6], Function[n, Times @@ Boole@ PrimeQ@ Map[First@ Reverse@ DeleteCases[Divisors@ #, d_ /; EvenQ@ d] &, n + Range[-1, 1]] == 1]] (* _Michael De Vlieger_, Aug 22 2016 *)
%t A276136 SequencePosition[Table[If[PrimeQ[Max[Select[Divisors[n],OddQ]]],1,0],{n,800000}],{1,1,1}][[;;,1]]+1 (* _Harvey P. Dale_, Jun 27 2023 *)
%o A276136 (Magma) [n: n in [2..3000000] | NumberOfDivisors(2*(n-1))- NumberOfDivisors(n-1)eq 2 and NumberOfDivisors(2(n))-NumberOfDivisors(n) eq 2 and NumberOfDivisors(2*(n+1))- NumberOfDivisors(n+1) eq 2];
%o A276136 (PARI) isA038550(n)=isprime(n>>valuation(n,2))
%o A276136 is(n)=isA038550(n-1) && isA038550(n) && isA038550(n+1) \\ _Charles R Greathouse IV_, Aug 22 2016
%o A276136 (PARI) forprime(p=2,1e11, my(a=isA038550(p-1),b=isA038550(p+1)); if(a && isA038550(p-2), print1(p-1", ")); if(a && b, print1(p", ")); if(b && isA038550(p+2), print1(p+1", "))) \\ may print numbers several times, but won't skip numbers; _Charles R Greathouse IV_, Aug 22 2016
%Y A276136 Supersequence of A181493. Subsequence of A038550.
%Y A276136 Cf. A001227, A181490, A181491, A181492, A275418.
%K A276136 nonn
%O A276136 1,1
%A A276136 _Juri-Stepan Gerasimov_, Aug 22 2016