A057897 Numbers which can be written as m^k-k, with m, k > 1.

2, 5, 7, 12, 14, 23, 24, 27, 34, 47, 58, 61, 62, 77, 79, 98, 119, 121, 122, 142, 167, 194, 213, 223, 238, 248, 252, 254, 287, 322, 340, 359, 398, 439, 482, 503, 509, 527, 574, 621, 623, 674, 723, 726, 727, 782, 839, 898, 959, 997, 1014, 1019, 1022, 1087, 1154

Offset: 1

Author

Henry Bottomley, Sep 26 2000

Keywords

• nonn

Comments

It may be that positive integers can be written as m^k-k (with m and k > 1) in at most one way [checked up to 10000].

All numbers < 10^16 of this form have a unique representation. The uniqueness question leads to a Pillai-like exponential Diophantine equation a^x-b^y = x-y for x > y > 1 and b > a > 1, which appears to have no solutions. - T. D. Noe, Oct 06 2004

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000325, A008865, A024024, A024037, A024050, A057895, A057898, A057899.

Cf. A099225 (numbers of the form m^k+k, with m and k > 1), A074981 (n such that there is no solution to Pillai's equation), A099226 (numbers that can be represented as both a^x+x and b^y-y, for some a, b, x, y > 1).

Programs

Mathematica

nLim=1000; lst={}; Do[k=2; While[n=m^k-k; n<=nLim, AppendTo[lst, n]; k++ ], {m, 2, Sqrt[nLim]}]; Union[lst] (* T. D. Noe, Oct 06 2004 *)

PARI

ok(n)={my(e=2); while(2^e <= n+e, if(ispower(n+e, e), return(1)); e++); 0} \\ Andrew Howroyd, Oct 20 2020

PARI

upto(lim)={my(p=logint(lim,2)); while(logint(lim+p+1,2)>p, p++); Vec(Set(concat(vector(p-1, e, e++; vector(sqrtnint(lim+e,e)-1, m, (m+1)^e-e)))))} \\ Andrew Howroyd, Oct 20 2020