A062316 Neither the sum or difference of 2 squares.

6, 14, 22, 30, 38, 42, 46, 54, 62, 66, 70, 78, 86, 94, 102, 110, 114, 118, 126, 134, 138, 142, 150, 154, 158, 166, 174, 182, 186, 190, 198, 206, 210, 214, 222, 230, 238, 246, 254, 258, 262, 266, 270, 278, 282, 286, 294, 302, 310, 318, 322, 326, 330, 334, 342, 350, 354, 358

Offset

1,1

Comments

Elements of A022544 congruent to 2 (mod 4).

Union of numbers congruent to 6 mod 8 (A017137) with numbers of the form 2 * A084109(n). - Franklin T. Adams-Watters, Jan 21 2007

Explanation: odd numbers are equal to the difference between two successive squares and among even numbers, multiples of 4 are of the form (k+2)^2-k^2, thus odd numbers and multiples of 4 are not in the sequence. Conversely, a difference of 2 squares cannot equal 2 (mod 4), thus this sequence contains the integers of the form 4k+2 that are in A022544 (not the sum of two squares); among integers of form 4k+2, this sequence contains all the integers of the form 8n+6 (A017137) that are not the sum of 2 squares because they have at least one prime factor congruent to 3 (mod 4) to an odd power; it also contains integers of the form 8n+2 = 2(4n+1) with 4n+1 not the sum of two squares, which is sequence A084109. - Jean-Christophe Hervé, Oct 24 2015

Links

Jean-Christophe Hervé, Table of n, a(n) for n = 1..2507

Formula

a(n) == 2 (mod 4). Subsequence of A016825 (non-differences of squares). All first differences are either 4 or 8, each of which occurs infinitely often. - David W. Wilson, Mar 09 2005

Lim_{n->inf} a(n)/n = 4.

Example

From Jean-Christophe Hervé, Oct 24 2015: (Start)
6, 14, 22, 30, 38, 46, ... are in the sequence because they equal 6 (mod 8).
42 = 2*3*7, 66 = 2*3*11, 114 = 2*7*11 are also in the sequence: of the form 2*(4n+1) with 4n+1 not the sum of 2 squares.
(End)

Maple

N:= 1000: # to get all terms <= N
S:= {seq(4*i+2, i=0..floor((N-2)/4))}
  minus {seq(seq(x^2 + y^2, y = x .. floor(sqrt(N-x^2)), 2), x=1..floor(sqrt(N)))}:
sort(convert(S, list)); # Robert Israel, Oct 25 2015

Mathematica

Select[Range@ 360, SquaresR[2, #] == 0 && Mod[#, 4] == 2 &] (* Michael De Vlieger, Oct 26 2015, after Harvey P. Dale at A022544 *)

Crossrefs

Cf. A022544, A016825, union of A017137 and 2*A084109, complement of A263715.

Cf. A097271, A079299.

Sequence in context: A023057 A197127 A197171 * A079299 A043445 A189785

Adjacent sequences: A062313 A062314 A062315 * A062317 A062318 A062319

Keyword

nonn

Author

Michel ten Voorde, Jul 05 2001

Extensions

More terms from David W. Wilson, Feb 11 2003

Status

approved