A016825 Positive integers congruent to 2 (mod 4): a(n) = 4*n+2, for n >= 0.

2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230, 234

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• nice

Comments

Twice the odd numbers, also called singly even numbers.

Numbers having equal numbers of odd and even divisors: A001227(a(n)) = A000005(2*a(n)). - Reinhard Zumkeller, Dec 28 2003

Continued fraction for coth(1/2) = (e+1)/(e-1). The continued fraction for tanh(1/2) = (e-1)/(e+1) would be a(0) = 0, a(n) = A016825(n-1), n >= 1.

No solutions to a(n) = b^2 - c^2. - Henry Bottomley, Jan 13 2001

Sequence gives m such that 8 is the largest power of 2 dividing A003629(k)^m-1 for any k. - Benoit Cloitre, Apr 05 2002

k such that Sum_{d|k} (-1)^d = A048272(k) = 0. - Benoit Cloitre, Apr 15 2002

Also k such that Sum_{d|k} phi(d)*mu(k/d) = A007431(k) = 0. - Benoit Cloitre, Apr 15 2002

Also k such that Sum_{d|k} (d/A000005(d))*mu(k/d) = 0, k such that Sum_{d|k}(A000005(d)/d)*mu(k/d) = 0. - Benoit Cloitre, Apr 19 2002

Solutions to phi(x) = phi(x/2); primorial numbers are here. - Labos Elemer, Dec 16 2002

Together with 1, numbers that are not the leg of a primitive Pythagorean triangle. - Lekraj Beedassy, Nov 25 2003

For n > 0: complement of A107750 and A023416(a(n)-1) = A023416(a(n)) <> A023416(a(n)+1). - Reinhard Zumkeller, May 23 2005

Also the minimal value of Sum_{i=1..n+2} (p(i) - p(i+1))^2, where p(n+3) = p(1), as p ranges over all permutations of {1,2,...,n+2} (see the Mihai reference). Example: a(2)=10 because the values of the sum for the permutations of {1,2,3,4} are 10 (8 times), 12 (8 times) and 18 (8 times). - Emeric Deutsch, Jul 30 2005

Except for a(n)=2, numbers having 4 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005

A139391(a(n)) = A006370(a(n)) = A005408(n). - Reinhard Zumkeller, Apr 17 2008

Also a(n) = (n-1) + n + (n+1) + (n+2), so a(n) and -a(n) are all the integers that are sums of four consecutive integers. - Rick L. Shepherd, Mar 21 2009

The denominator in Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - Mohammad K. Azarian, Oct 13 2011

This sequence gives the positive zeros of i^x + 1 = 0, x real, where i^x = exp(i*x*Pi/2). - Ilya Gutkovskiy, Aug 08 2015

Numbers k such that Sum_{j=1..k} j^3 is not a multiple of k. - Chai Wah Wu, Aug 23 2017

Numbers k such that Lucas(k) is a multiple of 3. - Bruno Berselli, Oct 17 2017

Also numbers k such that t^k == -1 (mod 5), where t is a term of A047221. - Bruno Berselli, Dec 28 2017

The even numbers form a ring, and these are the primes in that ring. Note that unique factorization into primes does not hold, since 60 = 2*30 = 6*10. - N. J. A. Sloane, Nov 11 2019

Also numbers ending with 10 in base 2. - John Keith, May 09 2022

Examples

			0.4621171572600097585023184... = 0 + 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))), i.e., c.f. for tanh(1/2).
2.1639534137386528487700040... = 2 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + ...)))), i.e., c.f. for coth(1/2).
		

References

• H. Bass, Mathematics, Mathematicians and Mathematics Education, Bull. Amer. Math. Soc. (N.S.) 42 (2004), no. 4, 417-430.
• Arthur Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
• J. R. Goldman, The Queen of Mathematics, 1998, p. 70.
• Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 262, 278.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Tanya Khovanova, Recursive Sequences
• D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
• I. Lukovits and D. Janezic, Enumeration of conjugated circuits in nanotubes, J. Chem. Inf. Comput. Sci. 44 (2004), 410-414.
• Vasile Mihai and Michael Woltermann, Problem 10725: The Smoothest and Roughest Permutations, Amer. Math. Monthly, 108 (March 2001), pp. 272-273.
• Paolo Emilio Ricci, Complex Spirals and Pseudo-Chebyshev Polynomials of Fractional Degree, Symmetry (2018) Vol. 10, No. 12, 671.
• William A. Stein, The modular forms database
• Eric Weisstein's World of Mathematics, Bishop Graph
• Eric Weisstein's World of Mathematics, Maximal Clique
• Eric Weisstein's World of Mathematics, Singly Even Number
• Eric Weisstein's World of Mathematics, Square Number
• G. Xiao, Contfrac
• Index entries for continued fractions for constants
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A107687, A154739, A285871.

First differences of A001105.

Cf. A160327 (decimal expansion).

Subsequence of A042963.

Essentially the complement of A042965.

Programs

GAP

Flat(List([0..70], n->4*n+2)); # Stefano Spezia, Jun 17 2019

Haskell

a016825 = (+ 2) . (* 4)
a016825_list = [2, 6 ..]  -- Reinhard Zumkeller, Feb 14 2012

Magma

[4*n+2 : n in [0..70]];

Maple

a := n -> 4*n+2: seq(a(n), n = 0 .. 70); # Stefano Spezia, Jun 17 2019

Mathematica

Range[2, 280, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
4*Range[0, 70] +2 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{2, -1}, {2, 6}, 70] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[2*(1+x)/(1-x)^2, {x,0,70}], x] (* Eric W. Weisstein, Dec 01 2017 *)
NestList[#+4&,2,60] (* Harvey P. Dale, Apr 08 2022 *)

PARI

a(n)= 4*n+2

PARI

contfrac(tanh(1/2)) \\ To illustrate the 3rd comment. - Harry J. Smith, May 09 2009 [Edited by M. F. Hasler, Mar 09 2020]

Sage

[4*n+2 for n in (0..70)] # G. C. Greubel, Jun 28 2019

Formula

a(n) = 4*n + 2, for n >= 0.

a(n) = 2*A005408(n). - Lekraj Beedassy, Nov 28 2003

a(n) = A118413(n+1,2) for n>1. - Reinhard Zumkeller, Apr 27 2006

From Michael Somos, Apr 11 2007: (Start)

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

E.g.f.: 2*(1+2*x)*exp(x).

a(n) = a(n-1) + 4.

a(-1-n) = -a(n). (End)

a(n) = 8*n - a(n-1) for n > 0, a(0)=2. - Vincenzo Librandi, Nov 20 2010

From Reinhard Zumkeller, Jun 11 2012, Jun 30 2012 and Jul 20 2012: (Start)

A080736(a(n)) = 0.

A007814(a(n)) = 1;

A037227(a(n)) = 3.

A214546(a(n)) = 0. (End)

a(n) = T(n+2) - T(n-2) where T(n) = n*(n+1)/2 = A000217(n). In general, if M(k,n) = 2*k*n + k, then M(k,n) = T(n+k) - T(n-k). - Charlie Marion, Feb 24 2020

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 1/sqrt(2-sqrt(2)) (A285871).

Product_{n>=1} (1 + (-1)^n/a(n)) = sqrt(1-1/sqrt(2)) (A154739). (End)