A308324 Numbers which can be written in the form m^k - m with m an odd prime and k a positive integer.

0, 6, 20, 24, 42, 78, 110, 120, 156, 240, 272, 336, 342, 506, 620, 726, 812, 930, 1320, 1332, 1640, 1806, 2162, 2184, 2394, 2756, 3120, 3422, 3660, 4422, 4896, 4970, 5256, 6162, 6558, 6806, 6840, 7832, 9312

Offset: 1

Author

Craig J. Beisel, May 20 2019

Keywords

• nonn
• easy

Comments

Besides the trivial example a(1)=0, the only known term which has two representations is a(24) = 2184 = 3^7 - 3 = 13^3 - 13. It is conjectured by Bennett to be the only term with this property.

Examples

			a(3) = 5^2 - 5 = 20.
		

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.

Crossrefs

Cf. A057895, A246068.

Programs

PARI

x=List([]);lim=10000;forprime(m=3,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]));