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A290812 Odd composite numbers n such that k^(n - 1) == 1 (mod n) and gcd(k^((n - 1)/2) - 1, n) = 1 for some integer k in the interval [2, sqrt(n) + 1]. 1
91, 247, 325, 343, 485, 703, 871, 901, 931, 949, 1099, 1111, 1157, 1247, 1261, 1271, 1387, 1445, 1525, 1649, 1765, 1807, 1891, 1975, 2047, 2059, 2071, 2117, 2501, 2701, 2863, 2871, 3277, 3281, 3365, 3589, 3845, 4069, 4141, 4187, 4291, 4371, 4411, 4525 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If the condition "odd composite numbers" in the definition is replaced by "odd numbers", then every odd prime number is in the sequence.

This is not a subsequence of A290543 (for example, 65683 is missing in A290543).

LINKS

Arkadiusz Wesolowski and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from A. Wesolowski)

Wikipedia, Pocklington primality test

Index entries for sequences related to pseudoprimes

EXAMPLE

91 is in the sequence because:

1) it is an odd composite number.

2) k^90 == 1 (mod 91) and gcd(k^45 - 1, 91) = 1 with k = 10 < sqrt(91) + 1.

MATHEMATICA

Select[Range[3, 4525, 2], Function[n, And[CompositeQ@ n, AnyTrue[Range[2, Sqrt[n] + 1], And[PowerMod[#, n - 1, n] == 1, CoprimeQ[#^((n - 1)/2) - 1, n]] &]]]] (* Michael De Vlieger, Aug 16 2017 *)

PROG

(MAGMA) lst:=[]; for n in [3..4525 by 2] do if not IsPrime(n) then for a in [2..Floor(Sqrt(n)+1)] do if Modexp(a, n-1, n) eq 1 and GCD(a^Truncate((n-1)/2)-1, n) eq 1 then Append(~lst, n); break; end if; end for; end if; end for; lst;

(PARI) is(n) = if(n > 1 && n%2==1 && !ispseudoprime(n), for(x=2, sqrt(n)+1, if(Mod(x, n)^(n-1)==1 && gcd(x^((n-1)/2)-1, n)==1, return(1)))); 0 \\ Felix Fröhlich, Aug 18 2017

CROSSREFS

Cf. A130569, A290543.

Sequence in context: A293648 A225909 A051973 * A000864 A224460 A020441

Adjacent sequences:  A290809 A290810 A290811 * A290813 A290814 A290815

KEYWORD

nonn

AUTHOR

Arkadiusz Wesolowski, Aug 11 2017

STATUS

approved

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Last modified October 15 17:24 EDT 2019. Contains 328037 sequences. (Running on oeis4.)