A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

0, 2, 6, 14, 20, 24, 30, 42, 62, 78, 110, 120, 126, 156, 240, 254, 272, 336, 342, 506, 510, 620, 726, 812, 930, 1022, 1320, 1332, 1640, 1806, 2046, 2162, 2184, 2394, 2756, 3120, 3422, 3660, 4094, 4422, 4896, 4970, 5256, 6162, 6558, 6806, 6840, 7832, 8190, 9312

Offset

1,2

Comments

The only known terms which have two representations where m is prime are 6 and 2184. It is conjectured by Bennett these are the only terms with this property.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.

Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33.

Example

a(9) = 2^6 - 2 = 62.
For the two terms known to have two representations we have a(3) = 6 = 2^3 - 2 = 3^2 - 3 and a(33)= 2184 = 3^7 - 3 = 13^3 - 13.

Maple

N:= 10^6; # to get all terms <= N
P:= select(isprime, [2, seq(i, i=3..floor((1+sqrt(1+4*N))/2), 2)]):
S:= {0, seq(seq(m^k-m, k=2..floor(log[m](N+m))), m=P)}:
sort(convert(S, list)); # Robert Israel, Aug 11 2019

Prog

(PARI) x=List([]); lim=10000; forprime(m=2, lim, for(k=1, 100, y=(m^k-m); if(y>lim, break, i=setsearch(x, y, 1); if(i>0, listinsert(x, y, i))))); for(i=1, #x, print(x[i]));
(PARI) isok(n) = {forprime(p=2, oo, my(keepk = 0); for (k=1, oo, if ((x=p^k - p) == n, return(1)); if (x > n, keepk = k; break); ); if (keepk == 2, break); ); } \\ Michel Marcus, Aug 06 2019

Crossrefs

Cf. A057895, A057896, A246068, A308324.

Subsequences: A000918 (2^n - 2), A036689 (p^2 - p), A058809 (3^n - 3), A178671 (5^n - 5).

Sequence in context: A215807 A140525 A189804 * A246068 A032643 A067664

Adjacent sequences: A308391 A308392 A308393 * A308395 A308396 A308397

Keyword

nonn,easy

Author

Craig J. Beisel, May 24 2019

Status

approved