A078201 -id:A078201 - cp's OEIS frontend

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

Amarnath Murthy, Nov 21 2002

sign

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)

			a(4) = 4^1 - 1^4 = 3, a(10) = 3^10 - 10^3 = 58049.

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.

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 *)

Corrected and extended by Ryan Propper, Jun 20 2005

More terms from Alexander Adamchuk, Oct 08 2006

A241427 Smallest prime of the form n^k - k^n for some k, or 0 if no such prime exists.

Original entry on oeis.org

7, 2, 3, 6102977801, 5, 79792265017612001, 7, 2486784401

Offset: 2

Derek Orr, Aug 08 2014

nonn

Conjecture: a(n) > 0 for all n not in A097764.
More terms in b-file. If n > 4 and in A097764, n^k - k^n is factorable and won't be prime.
a(17) > 17^7500 - 7500^17. See A239279.

Derek Orr, Table of n, a(n) for n = 2..16

Cf. A078201.

a(n)=k=1;if(n>4,forprime(p=1,100,if(ispower(n)&&ispower(n)%p==0&&n%p==0,return(0));if(n%p==n,break)));k=1;while(!ispseudoprime(n^k-k^n),k++);return(n^k-k^n)
vector(15, n, a(n+1))

cp's OEIS Frontend

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.

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