A099228 -id:A099228 - cp's OEIS frontend

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

T. D. Noe, Oct 06 2004

nonn

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

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

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.

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]

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

A247336 Numbers of the form m^k - k - 1 with k > 0 and m > 1.

Original entry on oeis.org

0, 1, 4, 6, 11, 13, 22, 23, 26, 33, 46, 57, 60, 61, 76, 78, 97, 118, 120, 121, 141, 166, 193, 212, 222, 237, 247, 251, 253, 286, 321, 339, 358, 397, 438, 481, 502, 508, 526, 573, 620, 622, 673, 722, 725, 726, 781, 838, 897, 958, 996, 1013, 1018, 1021, 1086, 1153

Offset: 1

Juri-Stepan Gerasimov, Sep 13 2014

Primes in a(n): 11, 13, 23, 61, 97, 193, 251, 397, 673, 1013, 1021, 1153, ...

			0 is in this sequence because 2^1 - 1 - 1 = 0,
1 is in this sequence because 2^2 - 2 - 1 = 1.

Cf. A057897, A099226, A099228.

a(n + 1) = A057897(n) - 1.

A344990 Primes of the form m^k +- k, with m,k > 1.

Original entry on oeis.org

2, 5, 7, 11, 23, 37, 47, 61, 67, 79, 83, 167, 223, 227, 359, 439, 443, 503, 509, 521, 727, 839, 997, 1019, 1087, 1091, 1223, 1367, 1523, 1847, 2027, 2207, 2399, 2741, 3023, 3251, 3719, 3967, 4093, 4099, 4759, 5039, 5623, 5927, 6553, 6563, 6569, 7919, 8179

Offset: 1

Luke Voyles, Jun 05 2021

nonn

			    2 =  2^2 - 2,
    5 =  2^3 - 3,
    7 =  3^2 - 2,
   11 =  3^2 + 2,
   23 =  5^2 - 2,
   37 =  2^5 + 5,
   47 =  7^2 - 2,
   61 =  4^3 - 3,
   67 =  4^3 + 3,
   79 =  9^2 - 2,
   83 =  9^2 + 2,
  167 = 13^2 - 2,
  223 = 15^2 - 2,
  227 = 15^2 + 2,
  359 = 19^2 - 2,
  439 = 21^2 - 2,
  443 = 21^2 + 2,
  503 =  2^9 - 9,
  509 =  8^3 - 3,
  521 =  2^9 + 9,
  ...

Union of A099227 and A099228.

cp's OEIS Frontend

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

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A247336 Numbers of the form m^k - k - 1 with k > 0 and m > 1.

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A344990 Primes of the form m^k +- k, with m,k > 1.

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