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 A078202 a(n) is the smallest prime of the form abs(n^k - k^n), the absolute difference between n^k and k^n, or -1 if no such prime exists. 3
 2, 7, 2, 3, 7, 5, 79, 7, 431, 58049, 8375575711, 11, 13055867207, 13, 94233563770233419658037661865757455268745312881861761180195872329157714108064193, -1, 130783, 17, 523927, 19, 2046526777460104549122039297254727662107009 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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) LINKS EXAMPLE a(4) = 4^1 - 1^4 = 3, a(10) = 3^10 - 10^3 = 58049. MATHEMATICA Do[k = 1; While[ !PrimeQ[Abs[n^k - k^n]], k++ ]; Print[Abs[n^k - k^n]], {n, 1, 14}] (* Ryan Propper, Jun 20 2005 *) CROSSREFS Cf. A078201. Cf. A123701 = Minimum number k such that A078202(n) = abs(n^k - k^n) is prime. Cf. A122735 = Smallest prime of the form (n^k - k^n) for k > 1. Sequence in context: A307671 A011401 A265647 * A183335 A196329 A196653 Adjacent sequences:  A078199 A078200 A078201 * A078203 A078204 A078205 KEYWORD sign AUTHOR Amarnath Murthy, Nov 21 2002 EXTENSIONS Corrected and extended by Ryan Propper, Jun 20 2005 More terms from Alexander Adamchuk, Oct 08 2006 STATUS approved

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Last modified December 6 18:16 EST 2021. Contains 349567 sequences. (Running on oeis4.)