A007310 -id:A007310 - cp's OEIS frontend

A113801 Numbers that are congruent to {1, 13} mod 14.

1, 13, 15, 27, 29, 41, 43, 55, 57, 69, 71, 83, 85, 97, 99, 111, 113, 125, 127, 139, 141, 153, 155, 167, 169, 181, 183, 195, 197, 209, 211, 223, 225, 237, 239, 251, 253, 265, 267, 279, 281, 293, 295, 307, 309, 321, 323, 335, 337, 349, 351, 363, 365, 377, 379

Offset: 1

Giovanni Teofilatto, Jan 22 2006

If 14k+1 is a perfect square..(0,12,16,52,60,120..) then the square root of 14k+1 = a(n) - Gary Detlefs, Feb 22 2010

More generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in our case, a(n)^2-1==0 (mod 14). Also a(n)^2-1==0 (mod 28). - Bruno Berselli, Oct 26 2010 - Nov 17 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A113802, A113803, A113804, A113805, A113806, A113807, A008589, A045472 (primes), A195145 (partial sums), A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887.

a113801 n = a113801_list !! (n-1)
a113801_list = 1 : 13 : map (+ 14) a113801_list
-- Reinhard Zumkeller, Jan 07 2012

LinearRecurrence[{1,1,-1},{1,13,15},60] (* or *) Select[Range[500], MemberQ[{1,13},Mod[#,14]]&] (* Harvey P. Dale, May 11 2011 *)

a(n)=n\2*14-(-1)^n \\ Charles R Greathouse IV, Sep 15 2015

a(n) = 14*(n-1)-a(n-1), n>1. - R. J. Mathar, Jan 30 2010
From Bruno Berselli, Oct 26 2010: (Start)
a(n) = -a(-n+1) = (14*n+5*(-1)^n-7)/2.
G.f.: x*(1+12*x+x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2)+14 for n>2.
a(n) = 14*A000217(n-1)+1 - 2*sum[i=1..n-1] a(i) for n>1. (End)
a(0)=1, a(1)=13, a(2)=15, a(n)=a(n-1)+a(n-2)-a(n-3). - Harvey P. Dale, May 11 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/14)*cot(Pi/14). - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + ((14*x - 7)*exp(x) + 5*exp(-x))/2. - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/14).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/14)*cosec(Pi/14). (End)

Corrected and extended by Giovanni Teofilatto, Nov 14 2008

Replaced the various formulas by a correct one - R. J. Mathar, Jan 30 2010

A297845 Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 90, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81

Offset: 1

Rémy Sigrist, Jan 10 2018

For any number n > 0, let f(n) be the polynomial in a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
This table has many similarities with A248601.
For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
Also, f(1) = 0 and f(2) = 1.
The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.
See A297473 for the main diagonal of T.
As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication (A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - Peter Munn, Mar 25 2020
From Peter Munn, Jun 16 2021: (Start)
The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.
Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.
There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.
There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively.
(End)

			Array T(n, k) begins:
  n\k|  1   2   3    4    5    6    7     8    9    10
  ---+------------------------------------------------
    1|  1   1   1    1    1    1    1     1    1     1  -> A000012
    2|  1   2   3    4    5    6    7     8    9    10  -> A000027
    3|  1   3   5    9    7   15   11    27   25    21  -> A003961
    4|  1   4   9   16   25   36   49    64   81   100  -> A000290
    5|  1   5   7   25   11   35   13   125   49    55  -> A357852
    6|  1   6  15   36   35   90   77   216  225   210  -> A191002
    7|  1   7  11   49   13   77   17   343  121    91
    8|  1   8  27   64  125  216  343   512  729  1000  -> A000578
    9|  1   9  25   81   49  225  121   729  625   441
   10|  1  10  21  100   55  210   91  1000  441   550
From _Peter Munn_, Jun 24 2021: (Start)
The encoding, n, of polynomials, f(n), that is used for the table is further described in A206284. Examples of encoded polynomials:
   n      f(n)        n           f(n)
   1         0       16              4
   2         1       17            x^6
   3         x       21        x^3 + x
   4         2       25           2x^2
   5       x^2       27             3x
   6     x + 1       35      x^3 + x^2
   7       x^3       36         2x + 2
   8         3       49           2x^3
   9        2x       55      x^4 + x^2
  10   x^2 + 1       64              6
  11       x^4       77      x^4 + x^3
  12     x + 2       81             4x
  13       x^5       90   x^2 + 2x + 1
  15   x^2 + x       91      x^5 + x^3
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..5050
Encyclopedia of Mathematics, Additive arithmetic function
Encyclopedia of Mathematics, Isomorphism
Eric Weisstein's World of Mathematics, Distributive
Eric Weisstein's World of Mathematics, Ring.
Wikipedia, Polynomial ring

Cf. A001221, A007913, A048720, A061395, A055396, A206284, A248601, A248663.
Row n: n=1: A000012, n=2: A000027, n=3: A003961, n=4: A000290, n=5: A357852, n=6: A191002, n=8: A000578.
Main diagonal: A297473.
Functions f satisfying f(T(n,k)) = f(n) * f(k): A001222, A048675 (and similarly, other rows of A104244), A195017.
Powers of k: k=3: A000040, k=4: A001146, k=5: A031368, k=6: A007188 (see also A066117), k=7: A031377, k=8: A023365, k=9: main diagonal of A329050.
Integers in the ideal of the related ring (see Jun 2021 comment) generated by S: S={3}: A005408, S={4}: A000290\{0}, S={4,3}: A003159, S={5}: A007310, S={5,4}: A339690, S={6}: A325698, S={6,4}: A028260, S={7}: A007775, S={8}: A000578\{0}, S={8,3}: A191257, S={8,6}: A332820, S={9}: A016754, S={10,4}: A340784, S={11}: A008364, S={12,8}: A145784, S={13}: A008365, S={15,4}: A345452, S={15,9}: A046337, S={16}: A000583\{0}, S={17}: A008366.
Equivalent sequence for polynomial composition: A326376.
Related binary operations: A003991, A306697/A059896, A329329/A059897.

T(n,k) = my (f=factor(n), p=apply(primepi, f[, 1]~), g=factor(k), q=apply(primepi, g[, 1]~)); prod (i=1, #p, prod(j=1, #q, prime(p[i]+q[j]-1)^(f[i, 2]*g[j, 2])))

T is completely multiplicative in both parameters:
- for any n > 0
- and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
- T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n + i - 1)^e_i.
For any m > 0, n > 0 and k > 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n, 2) = n (2 is an identity element for T),
- T(n, 2^i) = n^i for any i >= 0,
- T(n, 4) = n^2 (A000290),
- T(n, 8) = n^3 (A000578),
- T(n, 3) = A003961(n),
- T(n, 3^i) = A003961(n)^i for any i >= 0,
- T(n, 6) = A191002(n),
- A001221(T(n, k)) <= A001221(n) * A001221(k),
- A001222(T(n, k)) = A001222(n) * A001222(k),
- A055396(T(n, k)) = A055396(n) + A055396(k) - 1 when n > 1 and k > 1,
- A061395(T(n, k)) = A061395(n) + A061395(k) - 1 when n > 1 and k > 1,
- T(A000040(n), A000040(k)) = A000040(n + k - 1),
- T(A000040(n)^i, A000040(k)^j) = A000040(n + k - 1)^(i * j) for any i >= 0 and j >= 0.
From Peter Munn, Mar 13 2020 and Apr 20 2021: (Start)
T(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).
T(n, m*k) = T(n, m) * T(n, k); T(n*m, k) = T(n, k) * T(m, k) (T distributes over multiplication).
A104244(m, T(n, k)) = A104244(m, n) * A104244(m, k).
For example, for m = 2, the above formula is equivalent to A048675(T(n, k)) = A048675(n) * A048675(k).
A195017(T(n, k)) = A195017(n) * A195017(k).
A248663(T(n, k)) = A048720(A248663(n), A248663(k)).
T(n, k) = A306697(n, k) if and only if T(n, k) = A329329(n, k).
A007913(T(n, k)) = A007913(T(A007913(n), A007913(k))) = A007913(A329329(n, k)).
(End)

New name from Peter Munn, Jul 17 2021

A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

Original entry on oeis.org

1, 3, 6, 12, 18, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489, 501, 507, 519, 525, 537, 543, 555

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the deltoidal trihexagonal tiling, or the mta net.
In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020
This is one of the Laves tilings.

Colin Barker, Table of n, a(n) for n = 0..1000
Robert Ferreol, Pavage de diane et son dual
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Wikipedia, Laves tilings
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A008574, A298030 (partial sums), A298031 (for a tetravalent node), A298033 (hexavalent node), A306771.

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Join[{1, 3, 6, 12, 18}, LinearRecurrence[{1, 1, -1}, {33, 39, 51}, 60]] (* Jean-François Alcover, Jan 07 2019 *)
Join[{1,3,6,12,18},Table[If[EvenQ[n],9n-15,9n-12],{n,5,70}]] (* Harvey P. Dale, Aug 25 2019 *)

Vec((1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 9*x^5 - 3*x^7) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Theorem: For n >= 5, if n is even then a(n) = 9*n-15, otherwise a(n) = 9*n-12. The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. - N. J. A. Sloane, Jan 24 2018
G.f.: -(3*x^7 - 9*x^5 - 3*x^4 - 4*x^3 - 2*x^2 - 2*x - 1)/((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7. - Colin Barker, Jan 25 2018
a(n) = (3/2)*(6*n - (-1)^n - 9) for n>4. - Bruno Berselli, Jan 25 2018
a(n) = 3*A007310(n-1), n>4. - R. J. Mathar, Jan 29 2018

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

Jason Earls, Jul 14 2001

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

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

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

[(6*n*(n-1) + (2*n-1)*(-1)^n + 1)/4: n in [1..70]]; // Wesley Ivan Hurt, Apr 21 2021

seq(n^2+n+2*ceil(n/2)^2,n=0..48); # Gary Detlefs, Feb 23 2010

Select[Range[0, 3999], IntegerQ[Sqrt[6# + 1]] &] (* Harvey P. Dale, Mar 10 2013 *)

je=[]; for(n=0,7000, if(issquare(6*n+1),je=concat(je,n))); je

{ 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

def A062717(n): return (n*(3*n + 4) + 1 if n&1 else n*(3*n + 2))>>1 # Chai Wah Wu, Jan 31 2023

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)

A091999 Numbers that are congruent to {2, 10} mod 12.

Original entry on oeis.org

2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 122, 130, 134, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 266, 274, 278, 286, 290, 298, 302, 310, 314, 322, 326, 334

Offset: 1

Ray Chandler, Feb 21 2004

Numbers divisible by 2 but not by 3 or 4. - Robert Israel, Apr 24 2015
For n > 1, a(n) is representable as a sum of four but no fewer consecutive nonnegative integers, i.e., 10 = 1 + 2 + 3 + 4, 14 = 2 + 3 + 4 + 5, 22 = 4 + 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016
Essentially the same as A063221. - Omar E. Pol, Aug 16 2023

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Second row of A092260.
Cf. A109761 (subsequence).
Cf. A002193, A007310, A101263, A138591, A186424.
Cf. A017545, A017641, A063221.

a091999 n = a091999_list !! (n-1)
a091999_list = 2 : 10 : map (+ 12) a091999_list
-- Reinhard Zumkeller, Jan 21 2013

[6*n-3+(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Apr 23 2015

A091999:=n->6*n-3+(-1)^n: seq(A091999(n), n=1..100); # Wesley Ivan Hurt, Apr 23 2015

Flatten[#+{2,10}&/@(12*Range[0,30])] (* or *) LinearRecurrence[{1,1,-1},{2,10,14},60] (* Harvey P. Dale, Jun 24 2013 *)

a(n) = 6*n - 3 + (-1)^n \\ David Lovler, Jul 16 2022

a(n) = 2*A007310(n).
a(n) = A186424(n) - A186424(n-2), for n > 1.
a(n) = 12*(n-1) - a(n-1), with a(1)=2. - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = a(n-1) + a(n-2) - a(n-3); a(1)=2, a(2)=10, a(3)=14. - Harvey P. Dale, Jun 24 2013
a(n) = 6*n - 3 + (-1)^n. - Wesley Ivan Hurt, Apr 23 2015
E.g.f.: 2 + (6*x - 2)*cosh(x) + 2*(3*x - 2)*sinh(x). - Stefano Spezia, May 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)). - Amiram Eldar, Dec 13 2021
E.g.f.: 2 + (6*x - 3)*exp(x) + exp(-x). - David Lovler, Aug 08 2022
a(n) = A063221(n), n > 1. - Omar E. Pol, Aug 15 2023
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2) (A002193).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/12) (A101263). (End)

A036668 Hati numbers: of form 2^i3^jk, i+j even, (k,6)=1.

Original entry on oeis.org

1, 4, 5, 6, 7, 9, 11, 13, 16, 17, 19, 20, 23, 24, 25, 28, 29, 30, 31, 35, 36, 37, 41, 42, 43, 44, 45, 47, 49, 52, 53, 54, 55, 59, 61, 63, 64, 65, 66, 67, 68, 71, 73, 76, 77, 78, 79, 80, 81, 83, 85, 89, 91, 92, 95, 96, 97, 99, 100, 101, 102, 103, 107

Offset: 1

N. J. A. Sloane, Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

If n appears then 2n and 3n do not. - Benoit Cloitre, Jun 13 2002

Closed under multiplication. Each term is a product of a unique subset of {6} U A050376 \ {2,3}. - Peter Munn, Sep 14 2019

Robert Israel, Table of n, a(n) for n = 0..10000
Don McDonald, Obituary of Alan Robert Boyd, posted to sci.math Jan 02 1999; alternative link.

Cf. A003159, A007310, A014601, A036667, A050376, A052330, A325424 (complement), A325498 (first differences), A373136 (characteristic function).
Positions of 0's in A182582.
Subsequences: A084087, A339690, A352272, A352273.
Cf. also A121537, A145204, A225837, A307150, A329575, A334747, A334748, A339746.

N:= 1000: # to get all terms up to N
A:= {seq(2^i,i=0..ilog2(N))}:
Ae,Ao:= selectremove(issqr,A):
Be:= map(t -> seq(t*9^j, j=0 .. floor(log[9](N/t))),Ae):
Bo:= map(t -> seq(t*3*9^j,j=0..floor(log[9](N/(3*t)))),Ao):
B:= Be union Bo:
C1:= map(t -> seq(t*(6*i+1),i=0..floor((N/t -1)/6)),B):
C2:= map(t -> seq(t*(6*i+5),i=0..floor((N/t - 5)/6)),B):
A036668:= C1 union C2; # Robert Israel, May 09 2014

a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/2}],
IntegerQ]]] &]], {150}]; a  (* A036668 *)
(* Peter J. C. Moses, Apr 23 2019 *)

twos(n) = {local(r,m);r=0;m=n;while(m%2==0,m=m/2;r++);r}
threes(n) = {local(r,m);r=0;m=n;while(m%3==0,m=m/3;r++);r}
isA036668(n) = (twos(n)+threes(n))%2==0 \\ Michael B. Porter, Mar 16 2010

is(n)=(valuation(n,2)+valuation(n,3))%2==0 \\ Charles R Greathouse IV, Sep 10 2015

list(lim)=my(v=List(),N);for(n=0,logint(lim\=1,3),N=if(n%2,2*3^n,3^n); while(N<=lim, forstep(k=N,lim,[4*N,2*N], listput(v,k)); N<<=2)); Set(v) \\ Charles R Greathouse IV, Sep 10 2015

from itertools import count
def A036668(n):
    def f(x):
        c = n+x
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 28 2025

Formula

a(n) = 12/7 * n + O(log^2 n). - Charles R Greathouse IV, Sep 10 2015

{a(n)} = A052330({A014601(n)}), where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Sep 14 2019

Extensions

Offset changed by Chai Wah Wu, Jan 28 2025

A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).

Original entry on oeis.org

1, 5, 6, 7, 8, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 35, 36, 37, 40, 41, 42, 43, 47, 48, 49, 53, 55, 56, 59, 61, 64, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 101, 102, 103, 104, 107, 109, 113, 114, 115, 119, 121, 125, 127, 131, 133, 135

Offset: 1

Griffin N. Macris, Dec 15 2020

nonn

From Peter Munn, Mar 16 2021: (Start)
The positive integers in the multiplicative subgroup of the positive rationals generated by 8, 6, and A215848 (primes greater than 3).
This subgroup, denoted H, has two cosets: 2H = (1/3)H and 3H = (1/2)H. It follows that the sequence is one part of a 3-part partition of the positive integers with the property that each part's terms are half the even terms of one of the other parts and also one third of the multiples of 3 in the remaining part.
(End)
Positions of multiples of 3 in A276085 (and in A276075). Because A276085 is completely additive, this is closed under multiplication: if m and n are in the sequence then so is m*n. - Antti Karttunen, May 27 2024
The coset sequences mentioned in Peter Munn's comment above are A373261 and A373262. - Antti Karttunen, Jun 04 2024

Robert Israel, Table of n, a(n) for n = 1..10000

Sequences of positive integers in a multiplicative subgroup of positive rationals generated by a set S and A215848: S={}: A007310, S={6}: A064615, S={3,4}: A003159, S={2,9}: A007417, S={4,6}: A036668, S={3,8}: A191257, S={4,9}: A339690, S={6,8}: this sequence.
Positions of 0's in A373153, positions of multiples of 3 in A276085 and in A372576.
Cf. A372573 (characteristic function), A373261, A373262.
Subsequences: A064615, A373144, A373373, A373484, A373837, A374042, A374044, A374120, A377872.
Sequences giving positions of multiples of k in A276085, for k=2, 3, 4, 5, 8, 27, 3125: A003159, this sequence, A369002, A373140, A373138, A377872, A377878.
Cf. also A332820, A373992, A383288.

N:= 1000: # for terms <= N
R:= {}:
for k1 from 0 to floor(N/6) do
  for k0 in [1,5] do
    k:= k0 + 6*k1;
    for j from 0 while 3^j*k <= N do
      for i from (j mod 3) by 3 do
        x:= 2^i * 3^j * k;
        if x > N then break fi;
        R:= R union {x}
od od od od:
sort(convert(R,list)); # Robert Israel, Apr 08 2021

Select[Range[130], Mod[IntegerExponent[#, 2] - IntegerExponent[#, 3], 3] == 0 &]

isA339746 = A372573; \\ Antti Karttunen, Jun 04 2024

from sympy import factorint
def ok(n):
  f = factorint(n, limit=4)
  i, j = 0 if 2 not in f else f[2], 0 if 3 not in f else f[3]
  return (i-j)%3 == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(200)) # Michael S. Branicky, Mar 26 2021

from itertools import count
def A339746(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c -= (m-1)//6+(m-5)//6+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 12 2025

Formula

a(n) ~ (91/43)*n.

A038179 Result of second stage of sieve of Eratosthenes (after eliminating multiples of 2 and 3).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151

Offset: 1

N. J. A. Sloane, Dec 11 1999

2, 3 and numbers of the form 6m +- 1.
Apart from first two terms, same as A007310.
Terms of this sequence (starting from the third term) are equal to the result of the expression sqrt(4!*(k+1) + 1) - but only when this expression yields integral values (that is when the parameter k takes values, which are terms of A144065). - Alexander R. Povolotsky, Sep 09 2008
For every integer n>2, n is in this sequence iff Product_{k=2..oo} 1/(1 - 1/k^n) = Product_{k=1..n} Gamma( 2 - (-1)^(k*(1 + 1/n)) ). - Federico Provvedi, Nov 07 2024

Fred S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 256.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Ahmed Hamdy A. Diab, Sequence eliminating law (SEL) and the interval formulas of prime numbers, arXiv:2012.03052 [math.NT], 2020.
H. B. Meyer, Eratosthenes' sieve.
Index entries for sequences generated by sieves.

Cf. A004280, A007310, A144065.

max = 200; Complement[Range[2, max], 2Range[2, Ceiling[max/2]], 6Range[2, Ceiling[max/6]] + 3] (* Alonso del Arte, May 16 2014 *)
Prepend[Table[3*n - Mod[ Mod[n, 2] + 1, n], {n, 1, 999}], 2] (* Mikk Heidemaa, Nov 02 2017 *)

O.g.f.: x*(2 + x + x^3 + 2x^4)/((1+x)*(1-x)^2). - R. J. Mathar, May 23 2008
a(n) = (1/9)*(4*n^3 + 3*n^2 + 1 - Kronecker(-3,n+1)). - Ralf Stephan, Jun 01 2014
From Mikk Heidemaa, Oct 28 2017: (Start)
a(n) = floor((41/21 - (3 mod n))^(-3*n+5)) + 3*n - 4 (n > 0).
a(n+1) = 3*n - ((n mod 2)+1) mod n (n > 0). (End)
a(n+2) = 2*floor((3*n+1)/2) + 1 for n>=1; see (17) in Diab link. - Michel Marcus, Dec 14 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = (7-sqrt(3)*Pi)/6. - Amiram Eldar, Sep 22 2022

Name edited by Michel Marcus, Dec 14 2020

A077259 First member of the Diophantine pair (m,k) that satisfies 5*(m^2 + m) = k^2 + k; a(n) = m.

Original entry on oeis.org

0, 2, 6, 44, 116, 798, 2090, 14328, 37512, 257114, 673134, 4613732, 12078908, 82790070, 216747218, 1485607536, 3889371024, 26658145586, 69791931222, 478361013020, 1252365390980, 8583840088782, 22472785106426, 154030760585064, 403257766524696, 2763969850442378

Offset: 0

Bruce Corrigan (scentman(AT)myfamily.com), Nov 01 2002

			a(3) = (2*6) - 2 + (2*17) = 12 - 2 + 34 = 44.
G.f. = 2*x + 6*x^2 + 44*x^3 + 116*x^4 + 798*x^5 + 2090*x^6 + 14328*x^7 + ... - _Michael Somos_, Jul 15 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021.
Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2022.
Hermann Stamm-Wilbrandt, 6 interlaced bisections
Index entries for linear recurrences with constant coefficients, signature (1,18,-18,-1,1).

Cf. A007805, A077260, A077261, A077262.

Cf. A053141.

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!(2*x*(x+1)^2/((1-x)*(x^2-4*x-1)*(x^2+4*x-1)))); // G. C. Greubel, Jul 15 2018

f := gfun:-rectoproc({a(-2) = 2, a(-1) = 0, a(0) = 0, a(1) = 2, a(n) = 18*a(n - 2) - a(n - 4) + 8}, a(n), remember): map(f, [$ (0 .. 40)])[]; # Vladimir Pletser, Jul 24 2020

LinearRecurrence[{1, 18, -18, -1, 1}, {0, 2, 6, 44, 116}, 30] (* G. C. Greubel, Jul 15 2018 *)
a[ n_] := With[{m = Max[n, -1 - n]}, SeriesCoefficient[ 2 x (x + 1)^2 / ((1 - x) (x^2 - 4 x - 1) (x^2 + 4 x - 1)), {x, 0, m}]]; (* Michael Somos, Jul 15 2018 *)

my(x='x+O('x^30)); concat([0], Vec(2*x*(x+1)^2/((1-x)*(x^2-4*x-1)*(x^2+4*x-1)))) \\ G. C. Greubel, Jul 15 2018

Let b(n) be A007805(n). Then with a(0)=0, a(1)=2, a(2*n+2) = 2*a(2*n+1) - a(2*n) + 2*b(n), a(2*n+3) = 2*a(2*n+2) - a(2*n+1) + 2*b(n+1).
a(n) = (A000045(A007310(n+1))-1)/2. - Vladeta Jovovic, Nov 02 2002 [corrected by R. J. Mathar, Sep 16 2009]
a(0)=0, a(1)=2, a(n+2) = 4 + 9*a(n) + 2*sqrt(1 +20*a(n) +20*a(n)^2). - Herbert Kociemba, May 12 2008
a(0)=0, a(1)=2, a(2)=6, a(3)=44, a(n) = 18*a(n-2) - a(n-4) + 8. - Robert Phillips, Sep 01 2008
G.f.: 2*x*(1+x)^2/((1-x)*(1+4*x-x^2)*(1-4*x-x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jul 15 2018
a(2*n) = A049651(2*n); a(2*n+1) = A110679(2*n+1). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019
a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Jul 24 2020
From Vladimir Pletser, Feb 07 2021: (Start)
a(n) = ((5+sqrt(5))*(2+sqrt(5))^n + (5-sqrt(5))*(2-sqrt(5))^n)/20 - 1/2 for even n;
a(n) = ((5+3*sqrt(5))*(2+sqrt(5))^n + (5-3*sqrt(5))*(2-sqrt(5))^n)/20 - 1/2 for odd n. (End)

More terms from Colin Barker, Mar 23 2014

A091998 Numbers that are congruent to {1, 11} mod 12.

Original entry on oeis.org

1, 11, 13, 23, 25, 35, 37, 47, 49, 59, 61, 71, 73, 83, 85, 95, 97, 107, 109, 119, 121, 131, 133, 143, 145, 155, 157, 167, 169, 179, 181, 191, 193, 203, 205, 215, 217, 227, 229, 239, 241, 251, 253, 263, 265, 275, 277, 287, 289, 299, 301, 311, 313, 323, 325, 335

Offset: 1

Ray Chandler, Feb 21 2004

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h and n in A000027), then ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in our case, a(n)^2 - 1 == 0 (mod 12). Also a(n)^2 - 1 == 0 (mod 24).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

First row of A092260.
Cf. A175885 (n == 1 or 10 (mod 11)), A175886 (n == 1 or 12 (mod 13)).
Cf. A097933 (primes), A195143 (partial sums).
Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A113801, A175887, A188887.

a091998 n = a091998_list !! (n-1)
a091998_list = 1 : 11 : map (+ 12) a091998_list
-- Reinhard Zumkeller, Jan 07 2012

[ n: n in [1..350] | n mod 12 eq 1 or n mod 12 eq 11 ];

LinearRecurrence[{1,1,-1},{1,11,13},100] (* Harvey P. Dale, Jul 26 2017 *)

is(n)=n=n%12;n==11 || n==1 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 12*n - a(n-1) - 12 (with a(1)=1). - Vincenzo Librandi, Nov 16 2010
a(n) = 6*n + 2*(-1)^n - 3.
G.f.: x*(1+10*x+x^2)/((1+x)*(1-x)^2).
a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.
a(n) = a(n-2) + 12 for n > 2.
a(n) = 12*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2 + sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + (6*x - 3)*exp(x) + 2*exp(-x). - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2 + sqrt(3)) = 2*cos(Pi/12) (A188887).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/3)*cos(Pi/12). (End)

Formulae and comment added by Bruno Berselli, Nov 17 2010 - Nov 18 2010

cp's OEIS Frontend

A113801 Numbers that are congruent to {1, 13} mod 14.

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A297845 Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.

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A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

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

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A091999 Numbers that are congruent to {2, 10} mod 12.

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A036668 Hati numbers: of form 2^i*3^j*k, i+j even, (k,6)=1.

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A036668 Hati numbers: of form 2^i3^jk, i+j even, (k,6)=1.

A339746 Positive integers of the form 2^i3^jk, gcd(k,6)=1, and i == j (mod 3).