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A375490
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Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).
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0
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1541, 2465, 4961, 30857, 31697, 72041, 83333, 162401, 192713, 206981, 258017, 359369, 544541, 565001, 574397, 653333, 929633, 1018601, 1032533, 1133441, 1351601, 1373633, 1904033, 1953281, 2035661, 2797349, 2864501, 3264797, 3375041, 3554633, 3562361, 3636161
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OFFSET
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1,1
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COMMENTS
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Note that if b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4).
Conjecture: In general, if k > 1 is odd and b^((k-1)/2) == -(b/k) (mod k), then (b/k) = -1. Under this conjecture, this sequence is equivalent to "Numbers k == 5 (mod 12) such that 3^((k-1)/2) == 1 (mod k)".
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LINKS
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EXAMPLE
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1541 is a term because (3/1541) = -1, and 3^((1541-1)/2) == 1 (mod 1541).
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PROG
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(PARI) isA375490(k) = (k>1) && gcd(k, 6)==1 && Mod(3, k)^((k-1)/2)==-kronecker(3, k)
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CROSSREFS
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| b=2 | b=3 | b=5 |
-----------------------------------+-------------------+----------+---------+
-----------------------------------+-------------------+----------+---------+
-----------------------------------+-------------------+----------+---------+
b^((k-1)/2)==-(b/k) (mod k), | | | |
(b/k)=-1, b^((k-1)/2)==1 (mod k) | | | |
-----------------------------------+-------------------+----------+---------+
(union of first two) | | | |
-----------------------------------+-------------------+----------+---------+
(union of all three) | | | |
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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