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Search: id:A073103
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| A073103 |
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Number of non-congruent solutions to x^4 == 1 (mod n). |
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3
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| 1, 1, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 8, 8, 4, 2, 2, 8, 4, 2, 2, 8, 4, 4, 2, 4, 4, 8, 2, 8, 4, 4, 8, 4, 4, 2, 8, 16, 4, 4, 2, 4, 8, 2, 2, 16, 2, 4, 8, 8, 4, 2, 8, 8, 4, 4, 2, 16, 4, 2, 4, 8, 16, 4, 2, 8, 4, 8, 2, 8, 4, 4, 8, 4, 4, 8, 2, 32, 2, 4, 2, 8, 16, 2, 8, 8, 4, 8, 8, 4, 4, 2, 8, 16, 4, 2, 4, 8
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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a(n)=2*A060594 (n) for n=5,10,13,15,16,17,20,25,26,29,..This subsequence, which contains all the primes of form 4k+1, seems to be asymptotic to 2n.
Multiplicative with a(p^e) = p^min(e-1, 3) if p = 2, 4 if p == 1 (mod 4), 2 if p == 3 (mod 4). David W. Wilson (davidwwilson(AT)comcast.net) Jun 09, 2005.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
S. R. Finch, Quartic and Octic Characters Modulo n (arXiv:0907.4894) [From S. R. Finch (Steven.Finch(AT)inria.fr), Aug 12 2009]
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FORMULA
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sum(k=1, n, a(k)) seems to be asymptotic to C*n*Log(n) with C>1.4...(when sum(k=1, A060594(k)) is asymptotic to C/2*n*Log(n) )
In fact, sum(k=1, n, a(k)) is asymptotic to c*n*log(n)^2 where 2*c=0.190876... [From S. R. Finch (Steven.Finch(AT)inria.fr), Aug 12 2009]
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PROGRAM
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(PARI) a(n)=sum(i=1, n, if((i^4-1)%n, 0, 1))
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CROSSREFS
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Cf. A060594.
Sequence in context: A057000 A090047 A088200 this_sequence A069177 A077659 A087692
Adjacent sequences: A073100 A073101 A073102 this_sequence A073104 A073105 A073106
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KEYWORD
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easy,nonn,mult
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AUTHOR
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Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2002
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