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A140394 Numbers n, satisfying A055231(n+1) - A055231(n) = 1, and with n and n+1 not squarefree. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 14:54 EDT 2019. Contains 328345 sequences. (Running on oeis4.)