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A047221
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Numbers that are congruent to {2, 3} mod 5.
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24
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2, 3, 7, 8, 12, 13, 17, 18, 22, 23, 27, 28, 32, 33, 37, 38, 42, 43, 47, 48, 52, 53, 57, 58, 62, 63, 67, 68, 72, 73, 77, 78, 82, 83, 87, 88, 92, 93, 97, 98, 102, 103, 107, 108, 112, 113, 117, 118, 122, 123, 127, 128, 132, 133, 137, 138, 142, 143, 147, 148, 152, 153
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Theorem: if 5^((n-1)/2) = -1 (mod n) then n = 2 or 3 (mod 5) (see Crandall and Pomerance).
Start with 2. The next number, 3, can not be written as the sum of two of the previous terms. So 3 is in. 4=2+2, 5=2+3, 6=3+3, so these are not in. But you can not obtain 7, so the next term is 7. And so on. - Fabian Rothelius (fabian.rothelius(AT)telia.com), Mar 13 2001
Primitive roots of 5. The first differences are periodic: 1,4,1,4,1,4.... - Paolo P. Lava (paoloplava(AT)gmail.com), Feb 29 2008
Also numbers n such that n^2 = -1 (mod 5). [From Vincenzo Librandi, Aug 05 2010]
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REFERENCES
| R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 3.24, p. 154.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (1,1,-1).
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FORMULA
| a(n) = -3+(1/2)*Sum_{k=0..n}{5-3*(-1)^k}. - Paolo P. Lava (paoloplava(AT)gmail.com), Feb 29 2008
a(n) = 5*(n-1)-a(n-1) (with a(1)=2). [From Vincenzo Librandi, Aug 05 2010]
a(n) = (10*n-3*(-1)^n-5)/4.
G.f.: x*(2+x+2*x^2)/((1+x)*(1-x)^2).
a(n)^2 = 5*A118015(a(n))+4.
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MAPLE
| P:=proc(n, m) local a, i, ok; for i from 1 by 1 to n do if (i^(m-1) mod m)=1 then a:=1; ok:=1; while a<m-1 do if (i^a mod m)=1 then ok:=0; fi; a:=a+1; od; if ok=1 then print(i); fi; fi; od; end: P(100, 5); - Paolo P. Lava (paoloplava(AT)gmail.com), Feb 29 2008
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MATHEMATICA
| Select[Range[0, 200], MemberQ[{2, 3}, Mod[#, 5]] &] (* From Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)
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PROG
| (MAGMA) [ n : n in [1..165] | n mod 5 eq 2 or n mod 5 eq 3 ];
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CROSSREFS
| Cf. A118015 (floor(n^2/5)).
Sequence in context: A002274 A102664 A055053 * A032967 A111101 A184860
Adjacent sequences: A047218 A047219 A047220 * A047222 A047223 A047224
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Apr 08 2002
Closed formula, g.f. and link added by Bruno Berselli (berselli.bruno(AT)yahoo.it), Nov 28 2010
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