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A002189 Pseudo-squares: a(n) = the least nonsquare positive integer which is 1 mod 8 and is a (nonzero) quadratic residue modulo the first n odd primes.
(Formerly M5039 N2175 N2326)
17
17, 73, 241, 1009, 2641, 8089, 18001, 53881, 87481, 117049, 515761, 1083289, 3206641, 3818929, 9257329, 22000801, 48473881, 48473881, 175244281, 427733329, 427733329, 898716289, 2805544681, 2805544681, 2805544681 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
Michael A. Bender, R Chowdhury, A Conway, The I/O Complexity of Computing Prime Tables, In: Kranakis E., Navarro G., Chávez E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science, vol 9644. Springer, Berlin, Heidelberg. See Footnote 9.
D. H. Lehmer, A sieve problem on "pseudo-squares", Math. Tables Other Aids Comp., 8 (1954), 241-242.
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N2175 and N2326.).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. C. Williams and Jeffrey Shallit, Factoring integers before computers, pp. 481-531 of Mathematics of Computation 1943-1993 (Vancouver, 1993), Proc. Symp. Appl. Math., Vol. 48, Amer. Math. Soc. 1994.
Kjell Wooding and H. C. Williams, "Doubly-focused enumeration of pseudosquares and pseudocubes". Proceedings of the 7th International Algorithmic Number Theory Symposium (ANTS VII, 2006).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..73 (from Bernstein link)
D. H. Lehmer, A sieve problem on "pseudo-squares", Math. Tables Other Aids Comp., 8 (1954), 241-242. [Annotated scanned copy]
D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451 [Annotated scanned copy]
R. F. Lukes, C. D. Patterson and H. C. Williams, Some results on pseudosquares, Mathematics of Computation 65:213 (1996), pp. 361-372.
Eric Weisstein's World of Mathematics, Pseudosquare
EXAMPLE
a(0) = 17 since 1 + 8*0 and 1 + 8*1 are squares, 17 = 1 + 8*2 is not and the quadratic residue condition is satisfied vacuosly. - Michael Somos, Nov 24 2018
MATHEMATICA
a[n_] := a[n] = (pp = Prime[ Range[2, n+1]]; k = If[ n == 0, 9, a[n-1] - 8]; While[ True, k += 8; If[ ! IntegerQ[ Sqrt[k]] && If[ Scan[ If[ ! (JacobiSymbol[k, #] == 1 ), Return[ False]] & , pp], , False, True], Break[]]]; k); Table[ Print[ an = a[n]]; an, {n, 0, 24}] (* Jean-François Alcover, Sep 30 2011 *)
a[ n_] := If[ n < 0, 0, Module[{k = If[ n == 0, 9, a[n - 1] - 8]}, While[ True, If[! IntegerQ[Sqrt[k += 8]] && Do[ If[ JacobiSymbol[k, Prime[i]] != 1, Return @ 0], {i, 2, n + 1}] =!= 0, Return @ k]]]]; (* Michael Somos, Nov 24 2018 *)
PROG
(PARI) a(n)=n=prime(n+1); for(s=4, 1e9, forstep(k=(s^2+7)>>3<<3+1, s^2+2*s, 8, forprime(p=3, n, if(kronecker(k, p)<1, next(2))); return(k))) \\ Charles R Greathouse IV, Mar 29 2012
CROSSREFS
Sequence in context: A141972 A161735 A142648 * A002224 A096637 A201610
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
The PSAM reference gives a table through p = 223 (the b-file here has many more terms).
More terms from Don Reble, Nov 14 2006
Additional references from Charles R Greathouse IV, Oct 13 2008
STATUS
approved

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Last modified July 20 17:32 EDT 2024. Contains 374459 sequences. (Running on oeis4.)