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Odd numbers not of the form p + 2^m with p prime and m >= 0 for which the smallest k in A067760 such that n + 2^k is prime increases.
1

%I #12 Sep 08 2022 08:46:17

%S 1,127,251,1657,1777,1973,3181,21893,31951,50839,67607,138977

%N Odd numbers not of the form p + 2^m with p prime and m >= 0 for which the smallest k in A067760 such that n + 2^k is prime increases.

%C There exist de Polignac numbers n such that for all k >= 1 the numbers n + 2^k are composite. It is conjectured that 30666137 is the smallest such number.

%C a(13) >= 453143.

%o (Magma) lst:=[]; c:=0; for n in [1..31951 by 2] do m:=-1; repeat m+:=1; a:=n-2^m; until a lt 1 or IsPrime(a); if a lt 1 then k:=0; repeat k+:=1; b:=n+2^k; until IsPrime(b); if k gt c then Append(~lst, n); c:=k; end if; end if; end for; lst;

%Y Cf. A006285, A067760, A263644.

%Y A276496 gives the record values.

%K nonn,hard,more

%O 1,2

%A _Arkadiusz Wesolowski_, Sep 05 2016