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Numbers that are congruent to {2, 3} mod 5.
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%I #84 Feb 08 2024 07:12:44

%S 2,3,7,8,12,13,17,18,22,23,27,28,32,33,37,38,42,43,47,48,52,53,57,58,

%T 62,63,67,68,72,73,77,78,82,83,87,88,92,93,97,98,102,103,107,108,112,

%U 113,117,118,122,123,127,128,132,133,137,138,142,143,147,148,152,153

%N Numbers that are congruent to {2, 3} mod 5.

%C Theorem: if 5^((n-1)/2) = -1 (mod n) then n == 2 or 3 (mod 5) (see Crandall and Pomerance).

%C 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

%C Also numbers k such that k^2 == -1 (mod 5). - _Vincenzo Librandi_, Aug 05 2010

%C For any (t,s) < n, a(t)*a(s) != a(n) and a(t) - a(s) != a(n). - _Anders Hellström_, Jul 01 2015

%C These numbers appear in the product of a Rogers-Ramanujan identity. See A003106 also for references. - _Wolfdieter Lang_, Oct 29 2016

%D Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 3.24, p. 154.

%H N. J. A. Sloane, <a href="/A047221/b047221.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F a(n) = 5*(n-1) - a(n-1) (with a(1)=2). - _Vincenzo Librandi_, Aug 05 2010

%F a(n) = (10*n - 3*(-1)^n - 5)/4.

%F G.f.: x*(2+x+2*x^2)/((1+x)*(1-x)^2).

%F a(n)^2 = 5*A118015(a(n)) + 4.

%F a(n) = 5 * (floor(n-1)/2) + 3 - n mod 2. - _Reinhard Zumkeller_, Nov 27 2012

%F Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/5. - _Amiram Eldar_, Dec 07 2021

%F E.g.f.: 2 + ((5*x - 5/2)*exp(x) - (3/2)*exp(-x))/2. - _David Lovler_, Aug 23 2022

%t {2,3}+#&/@(5 Range[0,30])//Flatten (* _Harvey P. Dale_, Jan 22 2023 *)

%o (Magma) [ n : n in [1..165] | n mod 5 eq 2 or n mod 5 eq 3 ];

%o (Haskell)

%o a047221 n = 5 * ((n - 1) `div` 2) + 3 - n `mod` 2

%o a047221_list = 2 : 3 : map (+ 5) a047221_list

%o -- _Reinhard Zumkeller_, Nov 27 2012

%o (PARI) Vec(x*(2+x+2*x^2)/((1+x)*(1-x)^2) + O(x^80)) \\ _Michel Marcus_, Jun 30 2015

%Y Cf. A118015 (floor(n^2/5)).

%Y Cf. A003631 (primes).

%Y Partitions into: A003106, A219607.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 08 2002

%E Closed formula, g.f. and link added by _Bruno Berselli_, Nov 28 2010