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A076980 Leyland numbers: 3, together with numbers expressible as n^k + k^n nontrivially, i.e., n,k > 1 (to avoid n = (n-1)^1 + 1^(n-1)). 15
3, 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 6250, 7073, 8361, 16580, 18785, 20412, 23401, 32993, 60049, 65792, 69632, 93312, 94932, 131361, 178478, 262468, 268705, 397585, 423393, 524649, 533169 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Crandall & Pomerance refer to these numbers in reference to 2638^4405 + 4405^2638, which was then the largest known prime of this form. - Alonso del Arte, Apr 05 2006 [Comment amended by N. J. A. Sloane, Apr 06 2015]
Conjecture: For d > 11, 10^(d-1)+(d-1)^10 is the smallest (base ten) d-digit term. - Hans Havermann, May 21 2018
Conjecture from Zhi-Wei Sun, Feb 26 2022: (Start)
(i) For each n > 0, we have a(n) <= p+1 < a(n+1) for some prime p.
(ii) a(n) < p < a(n+1) for some prime p, except that the interval (a(5), a(6)) = (54, 57) contains no prime. (End)
A013499 \ {1} is the subsequence of terms of the form 2*n^n, n > 1. - Bernard Schott, Mar 26 2022
REFERENCES
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2005.
LINKS
Hans Havermann, Table of n, a(n) for n = 1..5000 (terms 1..1001 from T. D. Noe)
Wikipedia, Leyland number.
EXAMPLE
a(9) = 177 because we can write 177 = 2^7 + 7^2.
MAPLE
N:= 10^7: # to get all terms <= N
A:= {3}:
for n from 2 to floor(N^(1/2)) do
for k from 2 do
a:= n^k + k^n;
if a > N then break fi;
A:= A union {a};
od
od:
A; # if using Maple 11 or earlier, uncomment the next line
# sort(convert(A, list)); # Robert Israel, Apr 13 2015
MATHEMATICA
Take[Sort[Flatten[Table[x^y + y^x, {x, 2, 100}, {y, x, 100}]]], 42] (* Alonso del Arte, Apr 05 2006 *)
nn=10^50; n=1; Union[Reap[While[n++; num=2*n^n; num<nn, Sow[num]; k=n; While[k++; num=n^k+k^n; num<nn, Sow[num]]]][[2, 1]]]
CROSSREFS
Prime subset of this sequence, A094133.
Cf. A013499.
Sequence in context: A105163 A011850 A141422 * A293057 A294417 A001580
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Oct 23 2002
EXTENSIONS
More terms from Benoit Cloitre, Oct 24 2002
More terms from Alonso del Arte, Apr 05 2006
STATUS
approved

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Last modified June 13 08:31 EDT 2024. Contains 373383 sequences. (Running on oeis4.)