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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

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

Cf. A003631 (primes), A094214.

Partitions into: A003106, A219607.

Programs

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

Magma

[ n : n in [1..165] | n mod 5 eq 2 or n mod 5 eq 3 ];

Mathematica

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

PARI

Vec(x*(2+x+2*x^2)/((1+x)*(1-x)^2) + O(x^80)) \\ Michel Marcus, Jun 30 2015

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

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 1.

Product_{n>=1} (1 + (-1)^n/a(n)) = 1/phi (A094214). (End)

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