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

4

`%I #22 Aug 08 2023 14:47:44
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`%S 11,37,67,83,227,443,521,1091,1523,2027,3251,4099,6563,6569,9803,
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`%T 10651,11027,12323,13691,15131,17579,21611,29243,32771,32783,47963,
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`%U 50627,56171,59051,62003,65027,74531,88211,91811,95483,103043,119027,123203
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`%N Primes of the form m^k+k, with m and k > 1.
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`%C It appears that primes of this form are much less common than primes of the form m^k-k (A099228).
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`%C 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
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`%H Vincenzo Librandi and Charles R Greathouse IV, <a href="/A099227/b099227.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Librandi)
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`%t 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]
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`%o (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
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`%Y Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A093324 (least k such that n^k+k is prime).
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`%Y Cf. A099228.
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`%K nonn
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`%O 1,1
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`%A _T. D. Noe_, Oct 06 2004
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