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Search: id:A086251
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| A086251 |
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Number of primitive prime factors of 2^n-1. |
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+0
6
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| 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3
(list; graph; listen)
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OFFSET
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1,11
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COMMENT
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A prime factor of 2^n-1 is called primitive if it does not divide 2^r-1 for any r<n. Equivalently, p is a primitive prime factor of 2^n-1 if ord(2,p)=n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..500 (using data from A001265)
Eric Weisstein's World of Mathematics, Zsigmondy Theorem
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FORMULA
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a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
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EXAMPLE
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a(11) = 2 because 2^11-1 = 23*89 and both 23 and 89 have order 11.
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MATHEMATICA
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Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]
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CROSSREFS
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Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252.
Sequence in context: A043286 A043285 A146291 this_sequence A092931 A147300 A110503
Adjacent sequences: A086248 A086249 A086250 this_sequence A086252 A086253 A086254
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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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