A047221 Numbers that are congruent to {2, 3} mod 5.

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

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

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) = 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

Mathematica

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

Prog

(Magma) [ n : n in [1..165] | n mod 5 eq 2 or n mod 5 eq 3 ];
(Haskell)
a047221 n = 5 * ((n - 1) `div` 2) + 3 - n `mod` 2
a047221_list = 2 : 3 : map (+ 5) a047221_list
-- Reinhard Zumkeller, Nov 27 2012
(PARI) Vec(x*(2+x+2*x^2)/((1+x)*(1-x)^2) + O(x^80)) \\ Michel Marcus, Jun 30 2015

Crossrefs

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

Cf. A003631 (primes).

Partitions into: A003106, A219607.

Sequence in context: A102664 A055053 A233998 * A331078 A032967 A111101

Adjacent sequences: A047218 A047219 A047220 * A047222 A047223 A047224

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Closed formula, g.f. and link added by Bruno Berselli, Nov 28 2010

Status

approved