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%I #27 Oct 01 2019 04:00:07
%S 49,1681,18490,23762,39325,57121,182182,453962,656914,843637,1431125,
%T 1608574,1609674,1940449,2328482,2948406,3203050,3721549,5606230,
%U 6352825,8984002,10000165,13502254,19326874,19740249,21006589,26623750,35558770,38067925,46297822
%N Numbers n, satisfying A055231(n+1) - A055231(n) = 1, and with n and n+1 not squarefree.
%C 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.
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 49, p. 18, Ellipses, Paris 2008.
%e 49 is in the sequence because A055231(50) - A055231(49) = A055231(2*5^2) - A055231(7^2) = 2 - 1 = 1;
%e 18490 is in the sequence because A055231(18491) - A055231(18490) = A055231(11*41^2) -A055231(2*5*43^2) = 11 - 10 = 1.
%p isA013929 := proc(n)
%p n>3 and not numtheory[issqrfree](n) ;
%p end proc:
%p isA140394 := proc(n)
%p isA013929(n) and isA013929(n+1) and (A055231(n+1) -A055231(n) = 1) ;
%p end proc:
%p for n from 1 do
%p if isA140394(n) then
%p print(n);
%p end if;
%p end do: # _R. J. Mathar_, Dec 23 2011
%t 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 *)
%Y Cf. A007913, A013929, A055231, A068781.
%K nonn
%O 1,1
%A _Michel Lagneau_, Dec 19 2011
%E a(24)-a(30) from _Amiram Eldar_, Oct 01 2019
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