A245459 -id:A245459 - cp's OEIS frontend

A240173 Numbers n such that k^n - 2^k is not prime for any k.

1, 2, 13, 43, 45, 51, 53, 55, 57, 63, 72, 77, 81, 84, 85, 89, 93, 103, 108, 117, 121, 129, 147, 149, 151, 163, 171, 173, 177, 183, 191, 213, 229, 231, 239, 241, 250, 259, 261, 263, 273, 283, 286, 291, 321, 331, 333, 344, 345, 351, 353, 361, 373, 381, 390, 399

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2014

nonn

Numbers n such that A245459(n) = 0.

Amiram Eldar, Table of n, a(n) for n = 1..64

Cf. A245459.

import sympy
def a(n):
  k = 2
  count = 0
  while k**n > 2**k:
    if sympy.isprime(k**n-2**k):
      count += 1
    k += 1
  return count
n = 1
while n < 1000:
  if not a(n):
    print(n, end=', ')
  n += 1 # Derek Orr, Aug 02 2014

A245459(a(n)) = 0.

a(13)-a(34) from Derek Orr, Aug 02 2014

a(35) onwards from Amiram Eldar, Oct 03 2024

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

Original entry on oeis.org

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

Juri-Stepan Gerasimov, Aug 15 2014

nonn

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

			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.

Cf. A045575, A123206, A245459, A245643.

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 *)

a(n) >= A245459(n).

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

A246060 Number of primes of the form k^(n - m) - m^k where n > 2 and positive k, m.

Original entry on oeis.org

1, 1, 1, 5, 4, 5, 4, 7, 2, 8, 3, 10, 1, 12, 7, 13, 1, 11, 6, 13, 6, 19, 3, 12, 4, 17, 4, 10, 2, 18, 4, 15, 3, 21, 6, 14, 8, 18, 9, 23, 7, 9, 7, 21, 5, 13, 6, 22, 8, 16, 8, 24, 5, 22, 9, 12, 6, 26, 9, 26, 11, 27, 5, 30, 14, 34, 9, 23, 9, 48, 7, 11, 14, 37, 8, 32

Offset: 3

Juri-Stepan Gerasimov, Aug 23 2014

nonn

			a(3) = 1 because 2^(3 - 1) - 1^2 = 3 is prime with k = 2 and m = 1;
a(4) = 1 because 2^(4 - 1) - 1^2 = 7 is prime with k = 2 and m = 1;
a(5) = 1 because 3^(5 - 2) - 2^3 = 19 is prime with k = 3 and m = 2.

Cf. A242113, A245459.

a(n) = {my(k=2, q, v=List([])); if(ispseudoprime(q=2^(n-1)-1), listput(v, q)); while(k^(n-2)>2^k, if(ispseudoprime(q=k^(n-2)-2^k), listput(v, q)); k++); for(m=3, n-2, for(t=2, k-1, if(ispseudoprime(q=t^(n-m)-m^t), listput(v, q)))); #Set(v); } \\ Jinyuan Wang, Feb 24 2020

Definition and a(7) corrected by Colin Barker, Sep 01 2014

More terms from Jinyuan Wang, Feb 24 2020

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A240173 Numbers n such that k^n - 2^k is not prime for any k.

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

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A246060 Number of primes of the form k^(n - m) - m^k where n > 2 and positive k, m.

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