|
|
COMMENTS
|
If p is a prime then a(p+1) = p, with k = 1.
a(15) = 15^68 - 68^15, a 79-digit (certified) prime. a(16), if it exists, is greater than 16^39000 - 39000^16. a(17)..a(21) = 130783, 17, 523927, 19, 21^32 - 32^21 a(22), if it exists, is greater than 22^4000 - 4000^22. - Ryan Propper, Jun 20 2005
a(16) does not exist because 16^k - k^16 = (2^k + k^4)*(2^k - k^4)*(4^k + k^8) is composite for all k>0 except k = 16 when 16^k - k^16 = 0. - Alexander Adamchuk, Oct 04 2006
From Alexander Adamchuk, Oct 08 2006: (Start)
a(16) = -1. a(64) = -1. a(p+1) = p for prime p (note that corresponding k = 1). Corresponding minimum numbers k such that a(n) = Abs[n^k - k^n] are listed in A123701[n] = {3, 5, 1, 1, 2, 1, 2, 1, 2, 3, 8, 1, 6, 1, 68, -1, 2, 1, 2, 1, 32, 0, 60, 1, 12, 5, 0, 0, 98, 1, 42, 1, 0, 69, 6, 0, 0, 1, 0, 0, 60, 1, 32, 1, 44, 0, 110, 1, 24, 9, 2, 3, 2, 1, 0, 0, 0, 93, 0, 1, 180, 1, 88, -1, ...}, where k = -1 corresponds to a(n) = -1 and k = 0 corresponds to unknown a(n).
Currently a(n) is not known for n = {22, 27, 28, 33, 36, 37, 39, 40, 46, 55, 56, 57, 59, ...}.
a(11) = A122735(8) = 8^11 - 11^8 = 8375575711.
a(23),...,a(26) = {5054406430037885272981046135356839275715337535595096730028585410509132307928805601, 23, 953962166381085484825907807, 1490116119372884249}.
a(29),...,a(32) = {206539819953120274082671951780133190199874283596796371019530391490632157734921141966645648468156156063312771029604269179320472997337565971011273, 29, 433701716540983075324378476772415372611417595782401142359682753, 31}.
a(34),a(35) = {4699430983941716970028771656710732728232409636582667368874494198279899620725264856063216685987945059885543, 1719070799748422589190392551}.
a(38) = 37.
a(41),...,a(45) = {5848323709692443853597758618333177807096734261529545472862754750637561785400251641976844727314401, 41, 52656145834259929956933044695165193898922574867326768896079818367, 43, 84721522804414816904952398305908708011513455440403306207160333176150520399}. (End)
|
