A308394 - OEIS

%I #24 Aug 12 2019 02:24:59

%S 0,2,6,14,20,24,30,42,62,78,110,120,126,156,240,254,272,336,342,506,

%T 510,620,726,812,930,1022,1320,1332,1640,1806,2046,2162,2184,2394,

%U 2756,3120,3422,3660,4094,4422,4896,4970,5256,6162,6558,6806,6840,7832,8190,9312

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

%C 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.

%H Robert Israel, <a href="/A308394/b308394.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Bennett, <a href="https://doi.org/10.4153/CJM-2001-036-6">On some exponential equations of S. S. Pillai</a>, Canad. J. Math. 53 (2001), 897-922.

%H Dana Mackenzie, <a href="http://math.colgate.edu/~integers/s33/s33.Abstract.html">2184: An Absurd (and Adsurd) Tale</a>, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33.

%e a(9) = 2^6 - 2 = 62.

%e 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.

%p N:= 10^6; # to get all terms <= N

%p P:= select(isprime,[2,seq(i,i=3..floor((1+sqrt(1+4*N))/2),2)]):

%p S:= {0,seq(seq(m^k-m,k=2..floor(log[m](N+m))),m=P)}:

%p sort(convert(S,list)); # _Robert Israel_, Aug 11 2019

%o (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]));

%o (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

%Y Cf. A057895, A057896, A246068, A308324.

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

%K nonn,easy

%O 1,2

%A _Craig J. Beisel_, May 24 2019