A201914 -id:A201914 - cp's OEIS frontend

A057775 a(n) is the least prime p such that p-1 is divisible by 2^n and not by 2^(n+1).

2, 3, 5, 41, 17, 97, 193, 641, 257, 7681, 13313, 18433, 12289, 40961, 114689, 163841, 65537, 1179649, 786433, 5767169, 7340033, 23068673, 104857601, 377487361, 754974721, 167772161, 469762049, 2013265921, 3489660929, 12348030977, 3221225473, 75161927681

Offset: 0

Labos Elemer, Nov 02 2000

nonn

If we drop the requirement that p-1 must not be divisible by 2^(n+1), we get instead A035089, which is a nondecreasing sequence. - Jeppe Stig Nielsen, Aug 09 2015

			a(13) = 40961 = 1 + 8192*5 where the last term is divisible by the 13th power of 2 and 40961 is the smallest prime with that property.

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A000040, A006093, A035050, A035089, A057776, A126717, A201914.

f:= proc(n) local p;
  for p from 2^n+1 by 2^(n+1) do
    if isprime(p) then return p fi
  od
end proc:
map(f, [$0..100]); # Robert Israel, Aug 10 2015

Table[k = 1; While[p = k*2^n + 1; ! PrimeQ[p], k = k + 2]; p, {n, 0, 40}] (* T. D. Noe, Dec 27 2011 *)

a(n)=forstep(k=1,9e99,2,isprime((k<Jeppe Stig Nielsen, Aug 09 2015

a(n) = prime(A057776(n+1)). - Amiram Eldar, Mar 16 2025

More terms from Larry Reeves (larryr(AT)acm.org), Nov 03 2000

A307503 Least prime containing at least n consecutive 1's in its binary representation.

Original entry on oeis.org

2, 2, 3, 7, 31, 31, 127, 127, 1021, 3583, 4093, 6143, 8191, 8191, 81919, 131071, 131071, 131071, 524287, 524287, 4194301, 14680063, 16777213, 67108859, 536870909, 536870909, 536870909, 536870909, 2147483647, 2147483647, 2147483647, 2147483647, 21474836479

Offset: 0

John Mason, Apr 11 2019

For n > 0, a(n) = A000040(m) for the lowest m such that A090000(m) >= n.

a(n) = A087522(n) for n = 0 through 7, and in all other cases when a(n) is a base 2 repunit (Mersenne) prime.

			a(0) = 2, the smallest prime containing >= 0 1's.
a(1) = 2, the smallest prime containing >= 1 consecutive 1's.
a(2) = 3, the smallest prime containing >= 2 consecutive 1's.

Chai Wah Wu, Table of n, a(n) for n = 0..3314

Cf. A038374, A090000, A087522, A201914.

Cf. A090593 (with exactly n consecutive ones).

nbo(n)=if (n==0, return (0)); n>>=valuation(n, 2); if(n<2, return(n)); my(e=valuation(n+1, 2)); max(e, nbo(n>>e)); \\ A038374
a(n) = my(p=2); while(nbo(p) < n, p=nextprime(p+1)); p; \\ Michel Marcus, Apr 14 2019

a(n) <= A201914(n). - Rémy Sigrist, Apr 11 2019

a(n) = min_{k>=n} A090593(k). - Chai Wah Wu, Apr 26 2019

a(28)-a(32) from Chai Wah Wu, Apr 26 2019

cp's OEIS Frontend

A057775 a(n) is the least prime p such that p-1 is divisible by 2^n and not by 2^(n+1).

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