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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
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
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
Subsequences: A000918 (2^n - 2), A036689 (p^2 - p), A058809 (3^n - 3), A178671 (5^n - 5).
Sequence in context: A373437 A140525 A189804 * A246068 A032643 A067664
KEYWORD
nonn,easy
AUTHOR
Craig J. Beisel, May 24 2019
STATUS
approved