A188903 -id:A188903 - cp's OEIS frontend

A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 47, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 8, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 1

Eric Chen, Dec 14 2014

It is known that a(254602) = -1, because |509203-2^k| is always divisible by 3, 5, 7, 13, 17, or 241. a(1147) is the first unknown term.
a((A101036(n)+1)/2) = -1, so there are infinitely many n such that a(n) = -1.
a((A133122(n)+1)/2) = A096502((A133122(n)-1)/2).

			a(12) = 2 because 2*12-1 = 23 and that 23-2^1 = 21 is not prime but 23-2^2 = 19 is.
a(69) = 6 because 2*69-1 = 137, |137-2^k| is composite for k = 1, 2, 3, 4, 5 and prime for k = 6.
Even the smallest k can be also very large. For example, a(169) = 791.
a(1147) > 65536.

Jinyuan Wang, Table of n, a(n) for n = 1..1146

Cf. A006285, A096502, A133122, A188903.

Cf. A046067, A046069, A067760.

Table[k = 1; While[!PrimeQ[Abs[(2*n-1) - 2^k]], k++]; k, {n, 1, 1000}]

A252168(n)={ my(k=1); n=2*n-1; while(!ispseudoprime(abs(n-2^k)), k++); k }

a(19) corrected by Jinyuan Wang, Mar 25 2023

A276417 a(n) = least positive k such that (2*n + 1) - 2^k is prime, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 0, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 0, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 0

Arkadiusz Wesolowski, Sep 02 2016

nonn

a(n) = 1 iff n is in A006254. - Robert Israel, Sep 02 2016

For n > 1, a(n) = 0 iff 2n+1 is de Polignac number, A006285. - Thomas Ordowski, Apr 13 2017

			a(14) = 4 because (2*14 + 1) - 2^k is composite for k = 1, 2, 3 and prime for k = 4.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A006254, A006285, A101036, A133122, A188903, A232460, A252168.

lst:=[]; for n in [1..173 by 2] do k:=0; c:=k; repeat k+:=1; c+:=1; a:=n-2^k; until a lt 1 or IsPrime(a); if a lt 1 then Append(~lst, 0); else Append(~lst, c); end if; end for; lst;

f:= proc(n) local k;
     for k from 1 do
       if 2*n < 2^k then return 0
       elif isprime(2*n+1-2^k) then return k
       fi
     od
end proc:
map(f, [$0..100]); # Robert Israel, Sep 02 2016

Table[If[n <= 2, 0, k = 1; While[! PrimeQ[2 n + 1 - 2^k], k++]; k], {n, 0, 120}] (* Michael De Vlieger, Sep 03 2016 *)

a(n) = my(k=1); while(2^k < 2*n+1, if(ispseudoprime((2*n+1)-2^k), return(k)); k++); return(0) \\ Felix Fröhlich, Sep 02 2016

If A188903(n) >= 2, then a(n) = log_2(A188903(n)), otherwise a(n) = 0.

cp's OEIS Frontend

A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

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