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A217277 Numbers n such that n^2-9 is divisible by consecutive primes beginning with 2. 3
5, 9, 15, 21, 27, 33, 51, 57, 87, 93, 123, 147, 213, 297, 333, 483, 753, 1053, 1347, 2307, 2643, 3237, 4287, 6003, 10293, 12477, 14403, 18147, 26247, 58803, 74253, 739203, 1166883, 3801333 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Searched to 10^8. If this sequence is finite with 34 terms, then there is only one solution to m! + 9 = n^2: m=6, n=27.

Consider the Diophantine equations m^2 - 9 = k*(n^2 - 9), which lead to the Pell-like equations m^2 - k*n^2 = 9*(-k + 1) (1) where the only primes dividing k are in P = {2,3,5,7,11,13,17,19} and such that for every k there is at least one solution (m,n) that belongs to the sequence.

As an example, if k = 10 = 2*5, the 2 pairs of solutions of m^2 - 10n^2 = -81; (27,9) and (333,1053) belong to the sequence but the other solutions seem to not be P-smooth. If k = 30 = 2*3*5;  m^2 - 30n^2 = -261 and (147, 27) is a solution  belonging to the sequence, ...

If the infinitely many solutions of the Pell-like equations are never P-smooth, then this sequence is finite and there is a precise answer to the extended Brocard's problem: There are exactly 3 solutions to m!+t^2 = n^2 with t=1; 1 solution with t = 3; ... - Robin Garcia, Oct 01 2012

All terms beyond the first are 3 mod 6, since otherwise n^2 - 9 = 2^k for some k and so n - 3 and n + 3 are both powers of 2. For n > 7 all terms are +-3 mod 30, a consequence of Størmer's theorem. - Charles R Greathouse IV, Oct 01 2012

No further terms up to 10^10. - Charles R Greathouse IV, Oct 01 2012

LINKS

Table of n, a(n) for n=1..34.

EXAMPLE

5^2 - 9 = 2^4; 9^2 -9 = 2^3*3^2; 27^2 -9 = 2^4*3^2*5 = 6!; 87^2 - 9 = 2^3*3^3*5*7; 333^2 - 9 = 2^5*3^2*5*7*11.

PROG

(PARI) is(n)=my(m=n^2-9, t); forprime(p=2, , t=valuation(m, p); if(t, m/=p^t, return(m==1)))

print1(5); forstep(n=9, 1e6, 6, if(is(n), print1(", "n)))

\\ Charles R Greathouse IV, Oct 01 2012

(PARI) is(n)=my(f=factor(n^2-9)[, 1]~); f==primes(#f) \\ Charles R Greathouse IV, Oct 01 2012

CROSSREFS

Cf. A055932, A194099.

Sequence in context: A095948 A215721 A161536 * A087679 A179187 A075343

Adjacent sequences:  A217274 A217275 A217276 * A217278 A217279 A217280

KEYWORD

nonn

AUTHOR

Robin Garcia, Sep 29 2012

STATUS

approved

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Last modified February 20 15:02 EST 2018. Contains 299380 sequences. (Running on oeis4.)