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A001132
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Primes = +-1 mod 8.
(Formerly M4354 N1824)
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17
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7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Primes p such that 2 is a quadratic residue mod p.
Also primes p such that p divides 2^(p-1)/2 - 1. - Cino Hilliard (hillcino368(AT)gmail.com), Sep 04 2004
A001132 is exactly formed by the prime numbers of A118905 : in fact at first every prime p of A118905 is p=u^2-v^2+2uv, with for example u odd and v even so that : p-1=4u'(u'+1)-4v'(2u'+1-v') when u=2u'+1 and v=2v'. u'(u'+1) is even and v'(2u'+1-v') is always even. At second hand if p=8k+-1, p has the shape x^2-2y^2 ; letting u=x-y and v=y, comes p=(x-y)^2-y^2+2(x-y)y=u^2-v^2+2uv so p is a sum of the two legs of a pythagorean triangle. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 16 2008]
These are also the primes of form x^2-2y^2, excluding 2. See A038873. - Tito Piezas III (tpiezas(AT)gmail.com), Dec 28 2008
Primes p such that p^2 mod 48 = 1. [From Gary Detlefs, Dec 29 2011]
Primes in A047522. [Reinhard Zumkeller, Jan 07 2012]
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. Banderier, Calcul de (2/p)
Index entries for related sequences
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MAPLE
| seq(`if`(member(ithprime(n) mod 8, {1, 7}), ithprime(n), NULL), n=1..109); # Nathaniel Johnston, Jun 26 2011
for n from 1 to 600 do if (ithprime(n)^2 mod 48 = 1) then print(ithprime(n)) fi od. [From Gary Detlefs, Dec 29 2011]
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MATHEMATICA
| Select[Prime[Range[250]], MemberQ[{1, 7}, Mod[#, 8]]&] (* From Harvey P. Dale, Apr 29 2011 *)
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PROG
| (Haskell)
a001132 n = a001132_list !! (n-1)
a001132_list = [x | x <- a047522_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012
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CROSSREFS
| For primes p such that x^m = 2 (mod p) has a solution see A001132 (for m=2), A040028 (m=3), A040098 (m=4), A040159 (m=5), A040992 (m=6), A042966 (m=7), A045315 (m=8), A049596 (m=9), A049542 (m=10) - A049595 (m=63). Jeff Lagarias (lagarias(AT)umich.edu) points out that all these sequences are different, although this may not be apparent from looking just at the initial terms.
Agrees with A038873 except for initial term.
A118905 [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 16 2008]
Cf. A010051.
Sequence in context: A198441 A058529 A120681 * A165353 A048976 A088546
Adjacent sequences: A001129 A001130 A001131 * A001133 A001134 A001135
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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