A099227 Primes of the form m^k+k, with m and k > 1.

11, 37, 67, 83, 227, 443, 521, 1091, 1523, 2027, 3251, 4099, 6563, 6569, 9803, 10651, 11027, 12323, 13691, 15131, 17579, 21611, 29243, 32771, 32783, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203

Offset

1,1

Comments

It appears that primes of this form are much less common than primes of the form m^k-k (A099228).

As N increases, squares <= N outnumber all higher powers <= N by an increasingly wide margin, so the above observation is increasingly a consequence of the fact that primes of the form m^2 + 2 are less common than primes of the form m^2 - 2. Among numbers of these two forms, multiples of 3 make up 2/3 of the former, but none of the latter. - Jon E. Schoenfield, Jun 05 2021

Links

Vincenzo Librandi and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Librandi)

Mathematica

nLim=200000; lst={}; Do[k=2; While[n=m^k+k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Select[Union[lst], PrimeQ]

Prog

(PARI) list(lim)=my(v=List()); for(e=2, logint(lim\=1, 2), forstep(n=3-e%2, sqrtnint(lim-e, e), 2, my(t=n^e+e); if(isprime(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Jun 23 2023

Crossrefs

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A093324 (least k such that n^k+k is prime).

Cf. A099228.

Sequence in context: A072859 A125744 A116057 * A122728 A265767 A031381

Adjacent sequences: A099224 A099225 A099226 * A099228 A099229 A099230

Keyword

nonn

Author

T. D. Noe, Oct 06 2004

Status

approved