A057662 -id:A057662 - cp's OEIS frontend

A057673 Smallest prime p such that |2^n - p| is a prime.

3, 5, 2, 3, 3, 3, 3, 19, 5, 3, 3, 19, 3, 13, 3, 19, 17, 13, 5, 19, 3, 19, 3, 37, 3, 61, 5, 79, 89, 3, 41, 19, 5, 79, 41, 31, 5, 31, 107, 7, 167, 31, 11, 67, 17, 139, 167, 127, 59, 139, 71, 139, 47, 379, 53, 67, 5, 13, 137, 607, 107, 31, 167, 409, 59, 79, 5, 19, 23, 19, 71, 577, 107

Offset: 0

Labos Elemer, Oct 19 2000

nonn

The absolute value is relevant only for first two terms, 2^0-a(0) = 1-3 = -2, 2^1-a(1) = 2-5 = -3. According to Goldbach's conjecture, every even number > 2 is the sum of two primes, which implies that for all further terms, a(n) < 2^n. - M. F. Hasler, Jan 13 2011

			n=7, 2^n=128. The smallest terms subtracted from 128 resulting in a prime are 1,15,19,... Neither 1 nor 15 are primes but 19 is a prime. It gives 109=128-19, so a(n)=19.

Hugo Pfoertner, Table of n, a(n) for n = 0..5000

Analog of A056206. Cf. A056208, A057662.

f[n_] := Block[{p = 2}, While[! PrimeQ[2^n - p], p = NextPrime@ p]; p]; Array[f, 60, 0]

A057673(n)=forprime( p=1,default(primelimit), ispseudoprime(abs(2^n-p))& return(p))

Offset corrected and initial term added by M. F. Hasler, Jan 13 2011

A057676 Smallest prime q such that 2^prime(n) - q is prime.

Original entry on oeis.org

2, 3, 3, 19, 19, 13, 13, 19, 37, 3, 19, 31, 31, 67, 127, 379, 607, 31, 19, 577, 181, 67, 97, 31, 349, 619, 97, 919, 31, 211, 577, 181, 13, 397, 31, 829, 19, 577, 577, 103, 1669, 199, 19, 31, 439, 1021, 601, 1621, 2017, 733, 3, 199, 2113, 619, 1861, 1297, 241, 967

Offset: 1

Labos Elemer, Oct 19 2000

nonn

			For n = 4, prime(4) = 11, 2^11 = 2048, p2 = 2048-p1 is satisfied at first with prime p1 = 19 resulting in prime p2 = 2029, so a(4) = 19.
For n = 31, prime(31) = 127, p2 = 2^127-p1 is satisfied first with p1 = 577 and p2 = 170141183460469231731687303715884105151, so a(31) = 577.

Amiram Eldar, Table of n, a(n) for n = 1..560

Cf. A057662, A056206, A056208.

spq[n_]:=Module[{p=2,t=2^Prime[n]},While[!PrimeQ[t-p],p=NextPrime[p]];p]; Array[spq,60] (* Harvey P. Dale, Jul 13 2025 *)

a(n) = {my(p = 1 << prime(n), q = 2); while(!isprime(p - q), q = nextprime(q + 1)); q;} \\ Amiram Eldar, Feb 18 2025

Offset corrected by Amiram Eldar, Feb 18 2025

cp's OEIS Frontend

A057673 Smallest prime p such that |2^n - p| is a prime.

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