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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
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)
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
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