cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A062717 Numbers m such that 6*m+1 is a perfect square.

Original entry on oeis.org

0, 4, 8, 20, 28, 48, 60, 88, 104, 140, 160, 204, 228, 280, 308, 368, 400, 468, 504, 580, 620, 704, 748, 840, 888, 988, 1040, 1148, 1204, 1320, 1380, 1504, 1568, 1700, 1768, 1908, 1980, 2128, 2204, 2360, 2440, 2604, 2688, 2860, 2948, 3128, 3220, 3408, 3504
Offset: 1

Views

Author

Jason Earls, Jul 14 2001

Keywords

Comments

X values of solutions to the equation 6*X^3 + X^2 = Y^2. - Mohamed Bouhamida, Nov 06 2007
Arithmetic averages of the k triangular numbers 0, 1, 3, 6, ..., (k-1)*k/2 that take integer values. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009 [edited by Jon E. Schoenfield, Jan 10 2015]
Even terms in A186423; union of A033579 and A033580, A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
a(n) are integers produced by Sum_{i = 1..k-1} i*(k-i)/k, for some k > 0. Values for k are given by A007310 = sqrt(6*a(n)+1), the square roots of those perfect squares. - Richard R. Forberg, Feb 16 2015
Equivalently, numbers of the form 2*h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the sixth comment of A152749). - Bruno Berselli, Feb 02 2017

Links

Crossrefs

Equals 4 * A001318.
Cf. A005563, A046092, A001082, A002378, A036666.
Cf. A160757, A000217. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009
Cf. A007310.
Diagonal of array A323674. - Sally Myers Moite, Feb 03 2019

Programs

  • Magma
    [(6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4: n in [1..70]]; // Wesley Ivan Hurt, Apr 21 2021
  • Maple
    seq(n^2+n+2*ceil(n/2)^2,n=0..48); # Gary Detlefs, Feb 23 2010
  • Mathematica
    Select[Range[0, 3999], IntegerQ[Sqrt[6# + 1]] &] (* Harvey P. Dale, Mar 10 2013 *)
  • PARI
    je=[]; for(n=0,7000, if(issquare(6*n+1),je=concat(je,n))); je
  • PARI
    { n=0; for (m=0, 10^9, if (issquare(6*m + 1), write("b062717.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 09 2009
  • Python
    def A062717(n): return (n*(3*n + 4) + 1 if n&1 else n*(3*n + 2))>>1 # Chai Wah Wu, Jan 31 2023

Formula

G.f.: 4*x^2*(1 + x + x^2) / ( (1+x)^2*(1-x)^3 ).
a(2*k) = k*(6*k+2), a(2*k+1) = 6*k^2 + 10*k + 4. - Mohamed Bouhamida, Nov 06 2007
a(n) = n^2 - n + 2*ceiling((n-1)/2)^2. - Gary Detlefs, Feb 23 2010
From Bruno Berselli, Nov 28 2010: (Start)
a(n) = (6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4.
6*a(n) + 1 = A007310(n)^2. (End)
E.g.f.: (3*x^2*exp(x) - x*exp(-x) + sinh(x))/2. - Ilya Gutkovskiy, Jun 18 2016
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 21 2021
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=2} 1/a(n) = (9-sqrt(3)*Pi)/6.
Sum_{n>=2} (-1)^n/a(n) = 3*(log(3)-1)/2. (End)

A067611 Numbers of the form 6xy +- x +- y, where x, y are positive integers.

Original entry on oeis.org

4, 6, 8, 9, 11, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 29, 31, 34, 35, 36, 37, 39, 41, 42, 43, 44, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 92, 93, 94
Offset: 1

Views

Author

Jon Perry, Feb 01 2002

Keywords

Comments

Equivalently, numbers n such that either 6n-1 or 6n+1 is composite (or both are).
Numbers k such that 36*k^2 - 1 is not a product of twin primes. - Artur Jasinski, Dec 12 2007
Apart from initial zero, union of A046953 and A046954. - Reinhard Zumkeller, Jul 13 2014
From Bob Selcoe, Nov 18 2014: (Start)
Complementary sequence to A002822.
For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:
f == k (mod 6k+-1);
g == (5k-1) (mod 6k-1);
h == (5k+1) (mod 6k+1).
All numbers in classes g and h will be in this sequence; for class f, the quotient must be >= 1.
When determining which numbers are contained in this sequence, it is only necessary to evaluate f, g and h when the moduli are prime and the dividends are >= 2*k*(3*k - 1) (i.e., A033579(k)).
(End)
From Jason Kimberley, Oct 14 2015: (Start)
Numbers n such that A001222(A136017(n)) > 2.
The disjoint union of A060461, A121763, and A121765.
(End)
From Ralf Steiner, Aug 08 2018 (Start)
Conjecture 1: With u(k) = floor(k(k + 1)/4) one has A071538(a(u(k))*6) = a(u(k)) - u(k) + 1, for k >= 2 (u > 1).
Conjecture 2: In the interval [T(k-1)+1, T(k)], with T(k) = A000217(k), k >= 2, there exists at least one number that is not a member of the present sequence. (End)
Also: numbers of the form n*p +- round(p/6) with some positive integer n and prime p >= 5. [Proof available on demand.] - M. F. Hasler, Jun 25 2019

Examples

			4 = 6ab - a - b with a = 1, b = 1.
6 = 6ab + a - b or 6ab - a + b with a = 1, b = 1.
5 cannot be obtained by any values of a and b in 6ab - a - b, 6ab - a + b, 6ab + a - b or 6ab + a + b.

Links

Crossrefs

Cf. A002822, A010051, A037074, A046953, A046954, A060461, A070043, A070799, A121763, A121765, A136017, A136050, A071538 (pi_2).
Cf. A323674 (numbers 6xy +- x +- y including repetitions). - Sally Myers Moite, Jan 27 2019

Programs

  • GAP
    Filtered([1..120], k-> not IsPrime(6*k-1) or not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019
  • Haskell
    a067611 n = a067611_list !! (n-1)
a067611_list = map (`div` 6) $
   filter (\x -> a010051' (x-1) == 0 || a010051' (x+1) == 0) [6,12..]
-- Reinhard Zumkeller, Jul 13 2014
  • Magma
    [n: n in [1..100] | not IsPrime(6*n-1) or not IsPrime(6*n+1)]; // Vincenzo Librandi, Nov 19 2014
  • Maple
    filter:= n -> not isprime(6*n+1) or not isprime(6*n-1):
select(filter, [$1..1000]); # Robert Israel, Nov 18 2014
  • Mathematica
    Select[Range[100], !PrimeQ[6# - 1] || !PrimeQ[6# + 1] &]
Select[Range[100],AnyTrue[6#+{1,-1},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 05 2019 *)
  • PARI
    for(n=1, 1e2, if(!isprime(6*n+1) || !isprime(6*n-1), print1(n", "))) \\ Altug Alkan, Nov 10 2015
  • Sage
    [n for n in (1..120) if not is_prime(6*n-1) or not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Extensions

Edited by Robert G. Wilson v, Feb 05 2002
Edited by Dean Hickerson, May 07 2002
Showing 1-2 of 2 results.

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