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.

2, 7, 2, 3, 7, 5, 79, 7, 431, 58049, 8375575711, 11, 13055867207, 13, 94233563770233419658037661865757455268745312881861761180195872329157714108064193, -1, 130783, 17, 523927, 19, 2046526777460104549122039297254727662107009

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

Table of n, a(n) for n=1..21.

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