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A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 16 04:07 EDT 2021. Contains 345055 sequences. (Running on oeis4.)