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 A047221 Numbers that are congruent to {2, 3} mod 5. 36
 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; text; internal format)
 OFFSET 1,1 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, cannot 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 cannot obtain 7, so the next term is 7. And so on. - Fabian Rothelius, Mar 13 2001 Primitive roots of 5. The first differences are periodic: 1,4,1,4,1,4,.... - Paolo P. Lava, Feb 29 2008 Also numbers k such that k^2 == -1 (mod 5). - Vincenzo Librandi, Aug 05 2010 For any (t,s) < n, a(t)*a(s) != a(n) and a(t) - a(s) != a(n). - Anders Hellström, Jul 01 2015 These numbers appear in the product of a Rogers-Ramanujan identity. See A003106 also for references. - Wolfdieter Lang, Oct 29 2016 REFERENCES Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 3.24, p. 154. LINKS N. J. A. Sloane, Table of n, a(n) for n = 1..10000 Index entries for linear recurrences with constant coefficients, signature (1,1,-1). FORMULA a(n) = -3 + (1/2)*Sum_{k=0..n} (5 - 3*(-1)^k). - Paolo P. Lava, Feb 29 2008 a(n) = 5*(n-1) - a(n-1) (with a(1)=2). - 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. a(n) = 5 * (floor(n-1)/2) + 3 - n mod 2. - Reinhard Zumkeller, Nov 27 2012 Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/5. - Amiram Eldar, Dec 07 2021 E.g.f.: 2 + ((5*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - David Lovler, Aug 23 2022 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

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Last modified October 6 21:32 EDT 2022. Contains 357270 sequences. (Running on oeis4.)