A140394 Numbers n, satisfying A055231(n+1) - A055231(n) = 1, and with n and n+1 not squarefree.

49, 1681, 18490, 23762, 39325, 57121, 182182, 453962, 656914, 843637, 1431125, 1608574, 1609674, 1940449, 2328482, 2948406, 3203050, 3721549, 5606230, 6352825, 8984002, 10000165, 13502254, 19326874, 19740249, 21006589, 26623750, 35558770, 38067925, 46297822

Offset: 1

Author

Michel Lagneau, Dec 19 2011

Keywords

• nonn

Comments

There exists an infinite number of numbers that are divisible by a square and satisfy A055231(n+1) - A055231(n) = 1 because the Fermat-Pell equation 2x^2 - y^2 = 1 admits an infinite number of solutions.

Examples

			49 is in the sequence because A055231(50) - A055231(49) = A055231(2*5^2) - A055231(7^2) = 2 - 1 = 1;
18490 is in the sequence because A055231(18491) - A055231(18490) = A055231(11*41^2) -A055231(2*5*43^2)  = 11 - 10 = 1.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 49, p. 18, Ellipses, Paris 2008.

Crossrefs

Cf. A007913, A013929, A055231, A068781.

Programs

Maple

isA013929 := proc(n)
    n>3 and not numtheory[issqrfree](n) ;
end proc:
isA140394 := proc(n)
    isA013929(n) and isA013929(n+1) and (A055231(n+1) -A055231(n) = 1)  ;
end proc:
for n from 1 do
    if isA140394(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Dec 23 2011

Mathematica

rad[n_] := Times @@ First /@ FactorInteger[n]; pow[n_] := Denominator[n / rad[n]^2]; aQ[n_] := !SquareFreeQ[n] && !SquareFreeQ[n + 1] && pow[n + 1] - pow[n] == 1; Select[Range[10^6], aQ] (* Amiram Eldar, Oct 01 2019 *)

Extensions

a(24)-a(30) from Amiram Eldar, Oct 01 2019