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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..30.

Example

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.

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 *)

Crossrefs

Cf. A007913, A013929, A055231, A068781.

Sequence in context: A203500 A069327 A088068 * A008843 A145848 A014942

Adjacent sequences: A140391 A140392 A140393 * A140395 A140396 A140397

Keyword

nonn

Author

Michel Lagneau, Dec 19 2011

Extensions

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

Status

approved