A014210 Next prime after 2^n.

2, 3, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Except for a(1) = 3 instead of 2, a(n) is the least prime obtained as a binomial transform of n numbers. E.g. a(5) = (1,5,10,10,5,1).(1,1,1,1,1,6)= 37. - Amarnath Murthy, Nov 26 2003

a(n) is the smallest m for which m>(tau(m))^n, where tau(m) is the number of divisors of m. [Vladimir Shevelev, May 31 2010]

Equivalently, "Smallest prime > 2^n" while in A104080 it is "Smallest prime >= 2^n". The only difference is the 2nd term with a(1) = 3 and A104080(1) = 2. - Bernard Schott, Oct 30 2020

References

• J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 615 pp. 82 and 279, Ellipses, Paris, 2004. Warning : gives Sum_{k>=1} 1/A104080(k) = 0.7404...

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 0..1000

Crossrefs

See A203074 for another version.

Cf. A014211, A014234, A054321, A104080, A338475.

Programs

Maple

[ seq( nextprime( 2^i ),i=0..40) ];

Mathematica

NextPrime[ n_Integer] := (k = n + 1; While[ !PrimeQ[k], k++ ]; k); Table[ NextPrime[2^n], {n, 0, 35} ]
f[n_] := NextPrime[2^n]; Array[f, 30, 0] (* Robert G. Wilson v, Jun 05 2015 *)
NextPrime[2^Range[0,40]] (* Harvey P. Dale, Jun 22 2017 *)

PARI

a(n) = nextprime(2^n+1); \\ Michel Marcus, Oct 30 2020

Formula

Sum_{k>=0} 1/a(k) = A338475. - Bernard Schott, Oct 30 2020