A005818 -id:A005818 - cp's OEIS frontend

A144396 The odd numbers greater than 1.

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133

Offset: 1

Paul Curtz, Oct 03 2008

Last number of the n-th row of the triangle described in A142717.
If negated, these are also the values at local minima of the sequence A141620.
a(n) is the shortest leg of the n-th Pythagorean triple with consecutive longer leg and hypotenuse. The n-th such triple is given by (2n+1,2n^2+2n, 2n^2+2n+1), so that the longer legs are A046092(n) and the hypotenuses are A099776(n). - Ant King, Feb 10 2011
Numbers k such that the symmetric representation of sigma(k) has a pair of bars as its ends (cf. A237593). - Omar E. Pol, Sep 28 2018
Numbers k such that there is a prime knot with k crossings and braid index 2. (IS this true with "prime" removed?) - Charles R Greathouse IV, Feb 14 2023

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1)

Complement of A004275 and of A004277.
Cf. A000217, A002378, A005408, A007395, A046092, A099776, A237593, A254858.
Essentially the same as A140139, A130773, A062545, A020735, A005818.

List([3,5..200],n->n); # Muniru A Asiru, Sep 28 2018

a144396 = (+ 1) . (* 2)
a144396_list = [3, 5 ..] -- Reinhard Zumkeller, Feb 09 2015

seq(n,n=3..200,2); # Muniru A Asiru, Sep 28 2018

Range[3, 200, 2] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

a(n)=2*n+1 \\ Charles R Greathouse IV, Feb 14 2023

a(n) = A005408(n+1) = A000290(n+1) - A000290(n).
G.f.: x*(3-x)/(1-x)^2. - Jaume Oliver Lafont, Aug 30 2009
a(n) = A254858(n-1,2). - Reinhard Zumkeller, Feb 09 2015

Edited by R. J. Mathar, May 21 2009

A096910 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

3, 7, 9, 9, 11, 11, 13, 15, 15, 17, 17, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 31, 31, 31, 31, 33, 33, 33, 33, 33, 33, 33, 35, 35, 35, 35, 37, 37, 37, 37, 39, 39, 39, 39, 39, 39, 41, 41, 41, 41, 41, 43, 43, 43, 43, 43, 43, 45, 45, 45

Offset: 1

Ray Chandler, Aug 15 2004

nonn

Sequence with repetitions removed is A005818. - Ivan Neretin, May 24 2015

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907, A096908, A096909 (other components of the quadruple), A046086, A046087, A020882 (Pythagorean triples ordered in a similar way).

mx = 50; res = {}; Do[If[GCD[b, c, d] > 1, Continue[]]; If[IntegerQ[a = Sqrt[d^2 - b^2 - c^2]] && a > 0 && a <= b, AppendTo[res, {a, b, c, d}]], {d, mx}, {c, d}, {b, c}]; res[[All, 4]] (* Ivan Neretin, May 24 2015 *)

A005767 Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85

Offset: 1

N. J. A. Sloane, Ralph Peterson (ralphp(AT)library.nrl.navy.mil)

All numbers not equal to some 2^k or 5*2^k [Fraser and Gordon]. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006

T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.

T. D. Noe, Table of n, a(n) for n = 1..1000
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.

Complement of A094958. Cf. A169580, A000378, A000419, A000408.

For primitive solutions see A005818.

z=100;lst={};Do[a2=a^2;Do[b2=b^2;Do[c2=c^2;e2=a2+b2+c2;e=Sqrt[e2];If[IntegerQ[e]&&e<=z,AppendTo[lst,e]],{c,b,1,-1}],{b,a,1,-1}],{a,1,z}];Union@lst (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)

is(n)=if(n%5,n,n/5)==2^valuation(n,2) \\ Charles R Greathouse IV, Mar 12 2013

def A005767(n):
    def f(x): return n+x.bit_length()+(x//5).bit_length()
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

a(n) = n + 2*log_2(n) + O(1). - Charles R Greathouse IV, Sep 01 2015

A169580(n) = a(n)^2. - R. J. Mathar, Aug 15 2023

More terms from T. D. Noe, Mar 04 2010

A020742 Pisot sequence T(7,9).

Original entry on oeis.org

7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 0

David W. Wilson

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A005408, A020735. See A008776 for definitions of Pisot sequences.
Cf. A016825, A028560.
Essentially the same as A005818.

T[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2]]; Table[a[n], {n, 0, z}]]; T[7, 9, 66] (* Michael De Vlieger, Aug 08 2016 *)

pisotT(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]));
  a
}
pisotT(50, 7, 9) \\ Colin Barker, Aug 08 2016

a(n) = 2*n + 7.
a(n) = 2*a(n-1) - a(n-2).
From Elmo R. Oliveira, Oct 30 2024: (Start)
G.f.: (7 - 5*x)/(1 - x)^2.
E.g.f.: (7 + 2*x)*exp(x).
a(n) = A016825(n+3)/2 = A028560(n+1) - A028560(n). (End)

A100319 Even numbers m such that at least one of m-1 and m+1 is composite.

Original entry on oeis.org

8, 10, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 104, 106, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 140, 142, 144, 146, 148

Offset: 1

Rick L. Shepherd, Nov 13 2004

nonn

Subsequence of A100318. For each k >= 0, a(k+1) = a(k) + 2 unless a(k) + 1 and a(k) + 3 are twin primes, in which case a(k+1) = a(k) + 4 (as a(k) - 1 and a(k) + 5 are divisible by 3).

The even nonisolated primes(n+1). - Juri-Stepan Gerasimov, Nov 09 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A100318 (supersequence containing odd and even n), A045718 (n such that at least one of n-1 and n+1 is prime).
Cf. A167692(the even nonisolated nonprimes). - Juri-Stepan Gerasimov, Nov 09 2009
Cf. A005818, A038179, A007310, A038511, A025584.
Complement of A014574 (average of twin prime pairs) w.r.t. A005843 (even numbers), except for missing term 2.

Select[2*Range[100], CompositeQ[#-1] || CompositeQ[#+1] &]  (* G. C. Greubel, Mar 09 2019 *)

forstep(n=4,300,2,if(isprime(n-1)+isprime(n+1)<=1,print1(n,",")))

[n for n in (3..250) if mod(n,2)==0 and (is_prime(n-1) + is_prime(n+1)) < 2] # G. C. Greubel, Mar 09 2019

a(n) = A167692(n+1). - Juri-Stepan Gerasimov, Nov 09 2009

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A144396 The odd numbers greater than 1.

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A096910 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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A005767 Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0).

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A020742 Pisot sequence T(7,9).

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A100319 Even numbers m such that at least one of m-1 and m+1 is composite.

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