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
KEYWORD
nonn,easy
AUTHOR
Craig J. Beisel, May 24 2019
STATUS
approved