A122119 Least prime factor of 2^(2^n) + 5^(2^n).

7, 29, 641, 17, 97, 193, 274568286337, 257, 211457, 12289

Offset: 0

Author

Zak Seidov, Oct 19 2006

Keywords

• more
• nonn

Comments

a(11) = 5189633, a(12) > 10^9, a(13) > 10^9, a(14) = 26411009, a(15) = 65537, a(16) > 10^9, a(17) > 10^9, a(18) = 93847553, a(19) > 10^9, a(20) = 230686721. - Robert Price, May 01 2013

10^15 < a(10) <= 1431459829606245207173905329679749121. - Max Alekseyev, Jun 28 2013

a(n) == 1 (mod 2^(n+1)). - Max Alekseyev, Jun 23 2013

The sequence of highest powers of 2 dividing a(n)-1 is {0, 2, 7, 4, 5, 6, 7, 8, 9, 12, ?, 12, ?, ?, 16, 16, ?, ?, 19, ?, 22}. a(2)-1 = 640 = 5*2^7, a(3)-1 = 16 = 2^4, a(4)-1 = 96 = 3*2^5, a(5)-1 = 192 = 3*2^6, a(6)-1 = 274568286336 = 3*109*6559831*2^7, a(7)-1 = 256 = 2^8,.... - Robert Price, May 02 2013

a(12) = 5488091137, a(13) = 1364951041. - Chai Wah Wu, Jul 16 2019

All the known Fermat primes > 5 are in the sequence. a(22) = 167772161, a(26) = 1352914698241, a(28) = 11726871330817, a(29) = 3221225473, a(33) = 206158430209, a(38) = 19340409532579841, a(40) = 46179488366593, a(41) = 87930143896502273, a(46) = 19703248369745921, a(48) = 26747441136906797057, a(50) = 38280596832649217, a(57) = 639871435056800071681, a(66) = 1328165573307087716353, a(71) = 188894659314785808547841, a(75) = 441106808431887820120915969, a(77) = 1272917285561380533496310661121, a(83) = 380028249247497910327235837953, a(91) = 34662321099990647697175478273, a(107) = 48028745941447155563907091045285889, a(110) = 77884452878022414427957444938301441. - Chai Wah Wu, Oct 14 2019

Links

• FactorDB, Status of 2^(2^n)+5^(2^n).
• Mersenneforum, C680 of A122119, post by user swellman, 2013-06-17.

Crossrefs

Cf. A074600 (2^n + 5^n), A094475 (primes of form 2^n+5^n).

Programs

Mathematica

Table[FactorInteger[2^(2^n)+5^(2^n)][[1,1]],{n,0,7}] (* James C. McMahon, Oct 26 2024 *)