A242113 a(n) = number of primes of the form k^n - m^k where k > m > 0.

0, 1, 2, 6, 7, 2, 14, 7, 11, 10, 33, 10, 42, 35, 47, 39, 122, 22, 248, 113, 247, 236, 751, 75, 1268, 812, 1422, 1531, 4543, 87, 8669, 5750, 8884, 10983, 29084, 2274, 58841, 41242, 58030, 74646, 216647, 11656, 419147, 313237, 364925, 617742, 1576642, 75542, 3071839, 2299620

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 15 2014

Keywords

• nonn

Comments

It would be good to have a proof that a(n) is always finite. - N. J. A. Sloane, Sep 06 2014

Examples

			a(2) = 1 because  2^2 - 1^2 = 3 is prime;
a(3) = 2 because  2^3 - 1^2 = 7 is prime and 3^3 - 2^3 = 19 is prime, but 2^3 - 2^3 < 0, 5^3 - 2^5 = 93 is not prime, 5^3 - 2^7 = 215 is not prime, 9^3 - 2^9 = 217 is not prime, 11^3 - 2^11 < 0.
More generally, primes of the form k^r - m^k where  k > m > 0:
r = 2: 3;
r = 3: 7, 19;
r = 4: 7, 17, 73, 593, 2273, 20369;
r = 5: 7, 23, 31, 179, 58537, 1951811, 1986949;
r = 6: 4818617, 24006497;
r = 7: 7, 47, 79, 103, 127, 1137, 2179, 77101, 162287, 543607, 1706527, 9940951, 6069961193, 25365130463;
r = 8: 31, 6553, 141793, 49046209, 815722529, 16983038753, 499709542049;
r = 9: 71, 151, 223, 431, 463, 487, 503, 4521799, 133227103, 10604491181, 1175888158183;
r = 10: 4177, 37097, 58049, 58537, 1803001, 2486784401, 3486783889, 41426502825041, 819626139497153, 52458394747474721.
		

Crossrefs

Cf. A045575, A123206, A245459, A245643.

Programs

Mathematica

f[r_] := Length@ Rest@ Union@ Flatten@ Table[ If[ PrimeQ[k^r - m^k], k^r - m^k, 0], {k, 2, 10000000}, {m, Floor[k^(r/k)]}]; Do[ Print[ f[r]], {r, 2, 50}] (* Robert G. Wilson v, Aug 25 2014 *)

Formula

a(n) >= A245459(n).

Extensions

a(10)-a(50) from Robert G. Wilson v, Aug 25 2014