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Search: id:A086249
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| A086249 |
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Number of base-2 Fermat pseudoprimes x that have ord(2,x) = n. |
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3
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| 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 3, 1, 2, 1, 1, 0, 12, 4, 3, 0, 1, 1, 1, 1, 12, 1, 1, 4, 5, 1, 9, 4, 10, 8, 3, 4, 25, 0, 10, 11, 11, 4, 1, 4, 15, 4, 22, 1, 57, 0, 1, 4, 10, 1, 24, 1, 11, 1, 41, 4, 86, 4, 10, 25, 11, 0, 21, 4, 7, 4, 10, 1, 52, 1, 7, 10, 22, 0, 26, 11, 56, 1
(list; graph; listen)
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OFFSET
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1,22
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COMMENT
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A base-2 Fermat pseudoprime is a composite number x such that 2^x = 2 mod x. For such an x, ord(2,x) is the smallest positive integer m such that 2^m = 1 mod x. For a number x to have order n, it must be a factor of 2^n-1 and not be a factor of 2^r-1 for r<n. Sequence A086250 lists the smallest pseudoprime of order n. Note that there is no pseudoprime of order n when 2^n-1 is prime. However that does not explain why there are none for 12, 27, 49 and 77.
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LINKS
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R. G. E. Pinch, Pseudoprimes and their factors (FTP)
Eric Weisstein's World of Mathematics, Pseudoprime
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EXAMPLE
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a(10) = 1 there is only 1 pseudoprime, 341 = 11*31, having order 10; that is, 2^10 = 1 mod 341.
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MATHEMATICA
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Table[d=Divisors[2^n-1]; cnt=0; Do[m=d[[i]]; If[ !PrimeQ[m]&&PowerMod[2, m, m]==2&&MultiplicativeOrder[2, m]==n, cnt++ ], {i, Length[d]}]; cnt, {n, 100}]
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CROSSREFS
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Cf. A001567 (base-2 pseudoprimes), A086250.
Sequence in context: A054869 A056931 A139569 this_sequence A016569 A072801 A098872
Adjacent sequences: A086246 A086247 A086248 this_sequence A086250 A086251 A086252
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KEYWORD
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hard,nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jul 14 2003
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