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A020236 Strong pseudoprimes to base 10. 4
9, 91, 1729, 4187, 6533, 8149, 8401, 10001, 11111, 19201, 21931, 50851, 79003, 83119, 94139, 100001, 102173, 118301, 118957, 134863, 139231, 148417, 158497, 166499, 188191, 196651, 201917, 216001, 226273, 231337, 237169, 251251, 287809, 302177 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..203 from R. J. Mathar)
EXAMPLE
From Alonso del Arte, Aug 10 2018: (Start)
9 is a strong pseudoprime to base 10. It's not enough to check that 10^8 = 1 mod 9. Since 8 = 1 * 2^3, we also need to verify that 10 = 1 mod 9 and 10^2 = 1 mod 9 as well. Since these are both equal to 1, we see that 9 is indeed a strong pseudoprime to base 10.
91 is also a strong pseudoprime to base 10. Besides checking that 10^90 = 1 mod 91, since 90 = 45 * 2, we also check that 10^45 = -1 mod 91; the -1 is enough to satisfy the definition of a strong pseudoprime.
99 is a Fermat pseudoprime to base 10 (see A005939) but it is not a strong pseudoprime to base 10. Although 10^98 = 1 mod 99, since 98 = 49 * 2, we have to check 10^49 mod 99, and there we find not -1 nor 1 but 10. Therefore 99 is not in this sequence. (End)
MATHEMATICA
strongPseudoprimeQ[b_, n_] := Module[{rems = Table[PowerMod[b, (n - 1)/2^expo, n], {expo, 0, IntegerExponent[n - 1, 2]}]}, (rems[[-1]] == 1 || MemberQ[rems, n - 1]) && PowerMod[b, n - 1, n] == 1]; max = 5000; Select[Complement[Range[2, max], Prime[Range[PrimePi[max]]]], strongPseudoprimeQ[10, #] &] (* Alonso del Arte, Aug 10 2018 *)
CROSSREFS
Sequence in context: A286786 A123792 A022520 * A068624 A034666 A022505
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified July 25 01:41 EDT 2024. Contains 374585 sequences. (Running on oeis4.)