A122478 Minimum number k>0 such that ((2n+1)^k - (2n-1)^k)/2 is prime.

3, 13, 3, 3, 5, 3, 17, 3, 163, 3, 3, 109, 3, 13, 19, 5, 3, 3, 1879, 3, 13, 379, 7, 1531, 7, 5, 337, 5, 3, 61, 19, 3, 23, 3, 11, 16417, 163, 23, 3, 5, 3, 3, 3, 5

Offset: 1

Author

Alexander Adamchuk, Sep 14 2006, Sep 17 2006, Oct 07 2006

Keywords

• hard
• more
• nonn

Comments

All a(n) are primes. Corresponding minimal primes of the form ((2n+1)^k - (2n-1)^k)/2 are {13, 609554401, 109, 193, 51001, 433, 44937854708156010721, 769, ...}.

a(46)-a(49) are 17, 5, 31, 3. a(51)-a(61) are 109, 5, 7, 89, 13, 3, 31, 53, 5, 3, 5. a(63)-a(69) are 3, 7, 19, 5, 167, 163, 293. a(71)-a(74) are 3, 3407, 3, 3. a(76)-a(77) are 3, 19.

a(45), a(50), a(62), a(70), a(75) are currently unknown.

a(45) > 30000. - Max Alekseyev, May 18 2010

Crossrefs

Cf. A028491, A121877.

Programs

Mathematica

s={};Do[k=1;Until[PrimeQ[((2n+1)^k-(2n-1)^k)/2],k++];AppendTo[s,k] ,{n,35}];s (* James C. McMahon, Oct 27 2024 *)

Extensions

a(36)=16417 from Max Alekseyev, May 11 2010