A175901 Numbers k such that there exists a smaller number j such that j^2-1 has exactly the same set of distinct prime divisors as k^2-1.

7, 17, 19, 26, 31, 41, 49, 53, 55, 65, 71, 76, 97, 109, 127, 129, 161, 191, 197, 199, 209, 239, 241, 251, 271, 289, 295, 351, 391, 401, 433, 449, 485, 511, 575, 577, 626, 647, 649, 685, 701, 703, 721, 727, 799, 811, 881, 883, 901, 967, 989, 1025, 1055, 1079

Offset

1,1

Comments

From Robert Israel, Mar 19 2026: (Start)

A002315(k) is a term for k > 1. This is because A002315(k)^2 - 1 = 2 * (A001653(k)^2+1), A001653(k)^2-1 is even, and A002315(k) > A001653(k) for k > 1. In particular, the sequence is infinite. There are also infinite subsequences corresponding to other Pell-type equations, e.g. A028230(k) is a term when k > 1 and k == 0 or 1 (mod 3) because 4*(A001570(k)^2-1) = 3*(A028230(k)^2-1). (End)

Links

Robert Israel, Table of n, a(n) for n = 1..1500

Formula

a(1)=7 because set of prime divisors of 7^2-1 is the same as for 5^2-1. The set is {2,3}.

Maple

R:= NULL: V:= 'V':
for n from 2 to 2000 do
  v:= NumberTheory:-PrimeFactors(n^2-1);
if assigned(V[v]) then R:= R, n else V[v]:= n fi;
od:
R; # Robert Israel, Mar 19 2026

Mathematica

aa = {}; bb = {}; Do[k = n^2 - 1; c = FactorInteger[k]; b = {}; Do[AppendTo[b, c[[m]][[1]]], {m, 1, Length[c]}]; If[Position[aa, b] != {}, AppendTo[bb, n], AppendTo[aa, b]], {n, 2, 10000}]; bb (*Artur Jasinski*)

Prog

(PARI) isok(n) = {pfs = factor(n^2-1)[, 1]; for (k = 2, n-1, if (factor(k^2-1)[, 1] == pfs, return (1)); ); return (0); } \\ Michel Marcus, Nov 04 2013

Crossrefs

Cf. A001570, A001653, A002315, A028230, A175902 (for corresponding k)

Sequence in context: A084704 A392001 A198032 * A140566 A278785 A156005

Adjacent sequences: A175898 A175899 A175900 * A175902 A175903 A175904

Keyword

nonn

Author

Artur Jasinski, Oct 11 2010

Extensions

Edited by N. J. A. Sloane, Oct 14 2010

Status

approved