A078202 -id:A078202 - cp's OEIS frontend

A123701 Minimum number k > 0 such that abs(n^k - k^n) = A078202(n) is prime, or -1 if such k > 0 does not exist.

3, 5, 1, 1, 2, 1, 2, 1, 2, 3, 8, 1, 6, 1, 68, -1, 2, 1, 2, 1, 32

Offset: 1

Alexander Adamchuk, Oct 08 2006

A078202(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. A078202(n) = {2, 7, 2, 3, 7, 5, 79, 7, 431, 58049, 8375575711, 11, 13055867207, 13, 94233563770233419658037661865757455268745312881861761180195872329157714108064193, -1, 130783, 17, ...}. a(n) = -1 for n = {16,64,...} when A078202(n) = -1. a(n) = 1 for n = {3,4,6,8,12,14,18,20,...} = A008864(n) Primes + 1, when A078202(p+1) = p. Currently a(n) is not known for n = {22,28,33,36,37,39,40,46,55,56,57,59,...}. a(23)-a(27) = {60,1,12,5,-1}. a(29)-a(32) = {98,1,42,1}. a(34)-a(35) = {69,6}. a(38) = 1. a(41)-a(45) = {60,1,32,1,44}. a(47)-a(54) = {110,1,24,9,2,3,2,1}. a(58) = 93. a(60)-a(64) = {1,180,1,88,-1}.

Let x >= 2 and y >= 1 and k >= 1 and n == x^(xy). Then either (x,y,k) = (2,1,3) or (x,y,k) = (2,1,1) or abs(n^k - k^n) is composite. If we have (x,y) == (2,1), then n == 4, and we can check that a(4) == 1. Therefore, if n != 4 is a power of a number of the form x^x, then a(n) == -1. - Lucas A. Brown, Mar 25 2024

Cf. A078202, A008864, A122735.

f[n_] := Block[{k = If[EvenQ@n || n < 4, 1, 2]}; While[ ! PrimeQ@Abs[n^k - k^n], k += 2]; k] (* Robert G. Wilson v *)

A123206 Primes of the form x^y - y^x, for x,y > 1.

Original entry on oeis.org

7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, 9007199254738183, 79792265017612001, 1490116119372884249

Offset: 1

Alexander Adamchuk, Oct 04 2006

nonn

These are the primes in A045575, numbers of the form x^y - y^x, for x,y > 1. This includes all primes from A122735, smallest prime of the form (n^k - k^n) for k>1.

If y=1 was allowed, any prime p could be obtained for x=p+1. This motivates to consider sequence A243100 of primes of the form x^(y+1)-y^x. - M. F. Hasler, Aug 19 2014

			The primes 6102977801 and 1490116119372884249 are of the form 5^y-y^5 (for y=14 and y=26) and therefore members of this sequence. The next larger primes of this form would have y > 4500 and would be much too large to be included. - _M. F. Hasler_, Aug 19 2014

T. D. Noe, Table of n, a(n) for n=1..101 (terms < 10^400)
H. Lifchitz & R. Lifchitz, PRP of the form x^y-y^x on primenumbers.net.

Cf. A045575, A122735, A078202, A082754, A055651, A094133.
A163319 is the subsequences for fixed x=3, A243114 for x=6.
Cf. A072180, A109387, A117705, A117706, A128447, A128449, A128450, A128451, A122003, A128453, A128454.

N:= 10^100: # to get all terms <= N
A:= NULL:
for x from 2 while x^(x+1) - (x+1)^x <= N do
   for y from x+1 do
      z:= x^y - y^x;
      if z > N then break
      elif z > 0 and isprime(z) then A:=A, z;
      fi
od od:
{A}; # Robert Israel, Aug 29 2014

Take[Select[Intersection[Flatten[Table[Abs[x^y-y^x],{x,2,120},{y,2,120}]]],PrimeQ[ # ]&],25]
nn=10^50; n=1; t=Union[Reap[While[n++; k=n+1; num=Abs[n^k-k^n]; num0&&PrimeQ[#]&]],nn]] (* Harvey P. Dale, Nov 23 2013 *)

a=[];for(S=1,199,for(x=2,S-2,ispseudoprime(p=x^(y=S-x)-y^x)&&a=concat(a,p)));Set(a) \\ May be incomplete in the upper range of values, i.e., beyond a given S=x+y, a larger S may yield a smaller prime (for small x). - M. F. Hasler, Aug 19 2014

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A123701 Minimum number k > 0 such that abs(n^k - k^n) = A078202(n) is prime, or -1 if such k > 0 does not exist.

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A123206 Primes of the form x^y - y^x, for x,y > 1.

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