OFFSET
1,1
COMMENTS
The terms of this sequence are solutions y^m of the Diophantine equation a * (b^q - 1)/(b-1) = y^m with 1 < a < b, y >= 2, q >= 3, m >= 2. This equation has been studied by Kustaa A. Inkeri in Acta Arithmetica; some terms of this sequence come from his article where the author limits the study of this equation to bases b <= 100.
The case a = 1 is clarified in A208242; it corresponds to the Nagell-Ljunggren equation.
The sequence has infinitely many terms because the Diophantine equation 3*(x^2+x+1) = y^2 has infinitely many solutions. - Giovanni Resta, Apr 26 2019
The integers y such that y^2 (m = 2) satisfies this equation are in A158235, except 11 and 20 corresponding to a = 1. - Bernard Schott, Apr 27 2019
LINKS
Kustaa A. Inkeri, On the Diophantine equation a(x^n-1)/(x-1) = y^m, Acta Arithmetica, XXI, 1972.
EXAMPLE
3 * (22^3-1)/(22-1) = 39^2 and (333)_22 = 39^2 = 1521.
58 * (99^4-1)/(99-1) = 7540^2 and (AAAA)_99 = 7540^2 = 56851600 where A is the symbol for 58 in base 99.
MATHEMATICA
rupQ[n_, mx_] := Block[{t, x, p}, p = x^2 + x + 1; While[(t = p /. x -> mx) <= n && Reduce[p == n && x >= mx, x, Integers] === False, p = x*p + 1]; t <= n]; repdQ[n_] := AnyTrue[ Rest@ Most@ Divisors@ n, rupQ[n/#, #+1] &]; ex = 2; up = 10^7; L = {}; While[2^ex <= up, L = Union[L, Parallelize@ Select[ Range[2, Floor[ up^(1/ex)] ]^ex, repdQ]]; ex = NextPrime@ ex]; L (* Giovanni Resta, Apr 27 2019 *)
PROG
(PARI) isokb(n) = for(b=2, n-2, d=digits(n, b); if((#d > 2) && (vecmin(d)==vecmax(d)) && (d[1] > 1), return (1))); 0;
isok(n) = ispower(n) && isokb(n); \\ Michel Marcus, Apr 28 2019
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Bernard Schott, Apr 26 2019
EXTENSIONS
More terms from Giovanni Resta, Apr 26 2019
STATUS
approved