A068231 -id:A068231 - cp's OEIS frontend

A132241 Twin primes congruent to {11, 13} mod 30.

11, 13, 41, 43, 71, 73, 101, 103, 191, 193, 281, 283, 311, 313, 431, 433, 461, 463, 521, 523, 641, 643, 821, 823, 881, 883, 1031, 1033, 1061, 1063, 1091, 1093, 1151, 1153, 1301, 1303, 1451, 1453, 1481, 1483, 1721, 1723, 1871, 1873

Offset: 1

Omar E. Pol, Aug 15 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
C. K. Caldwell, Twin Primes.

Cf. A000040, A001097, A001359, A006512, A039949, A068231.

a[0]=1;a[n_]:=a[n]=a[n-1]+10;Flatten[Table[If[PrimeQ[a[n]]&&PrimeQ[a[n]+2],{a[n],a[n]+2},{}],{n,0,200}]] (* Vincenzo Librandi, Aug 15 2012 *)

A167057 Numbers k such that 12*k + 11 is prime.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 10, 13, 14, 15, 18, 19, 20, 21, 25, 28, 29, 31, 34, 35, 36, 38, 39, 40, 41, 46, 48, 49, 53, 54, 56, 59, 61, 68, 69, 71, 73, 75, 78, 80, 81, 84, 85, 90, 91, 95, 96, 98, 101, 104, 106, 108, 109, 113, 118, 119, 120, 123, 124, 125, 126, 129, 130, 131, 133

Offset: 1

Michael B. Porter, Oct 27 2009

Corresponds to even numbers in A024898.

			3 is in the sequence since 12*3+11 = 47 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A110801, A167055, A167056, A024898, primes are in A068231.

[n: n in [0..200] |IsPrime(12*n+11)]; // Vincenzo Librandi, Mar 25 2010

Select[Range[0, 200], PrimeQ[12 # + 11] &] (* Vincenzo Librandi, May 20 2014 *)

PARI
```
isA167057(n) = isprime(12*n+11)
```

a(n) = A138620(n)-1. [From R. J. Mathar, Oct 29 2009]

A014557 Multiplicity of K_3 in K_n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 4, 8, 12, 20, 28, 40, 52, 70, 88, 112, 136, 168, 200, 240, 280, 330, 380, 440, 500, 572, 644, 728, 812, 910, 1008, 1120, 1232, 1360, 1488, 1632, 1776, 1938, 2100, 2280, 2460, 2660, 2860, 3080, 3300, 3542, 3784, 4048, 4312, 4600, 4888, 5200

Offset: 0

Eric W. Weisstein

The multiplicity of triangles in K_n is defined to be the minimum number of monochromatic copies of K_3 that occur in any 2-coloring of the edges of K_n. - Allan Bickle, Mar 04 2023
Twice A008804 (up to offset).
From Alexander Adamchuk, Nov 29 2006: (Start)
n divides a(n) for n = {1,2,3,4,5,8,10,13,14,16,17,20,22,25,26,28,29,32,34,37,38,40,41,44,46,49,50,52,53,56,58,61,62,64,65,68,70,73,74,76,77,80,82,85,86,88,89,92,94,97,98,100,...}.
Prime p divides a(p) for p = {2,3,5,13,17,29,37,41,53,61,73,89,97,101,109,113,137,149,157,173,181,193,197,...} = (2,3) and all primes from A002144: Pythagorean primes: primes of form 4n+1.
(n+1) divides a(n) for n = {1,2,3,4,5,19,27,43,51,67,75,91,99,...}.
(p+1) divides a(p) for prime p = {2,3,5,19,43,67,139,163,211,283,307,331,379,499,523,547,571,619,643,691,739,787,811,859,883,907,...} = {2,5} and all primes from A141373: Primes of the form 3x^2+16y^2.
(n-1) divides a(n) for n = {2,3,4,5,21,29,45,53,69,77,93,101,...}.
(p-1) divides a(p) for prime p = {2,3,5,29,53,101,149,173,197,269,293,317,389,461,509,557,653,677,701,773,797,821,941,..} = {2,3} and all primes from A107003: Primes of the form 5x^2+2xy+5y^2, with x and y any integer.
(n-2) divides a(n) for n = {3,4,5,12,16,24,28,36,40,48,52,60,64,72,76,84,88,96,100,...} = {3,5} and 4*A032766: Numbers congruent to 0 or 1 mod 3.
(n+3) divides a(n) for n = {1,2,3,4,5,9,11,18,32,39}.
(n-3) divides a(n) for n = {4,5,7,9,23,31,47,55,71,79,95,103,119,127,143,151,167,175,...}.
(p+3) divides a(p) for prime p = {5,7,23,31,47,71,79,103,127,151,167,191,199,...} = {5} and all primes from A007522: Primes of form 8n+7.
(n-4) divides a(n) for n = {5,6,8,11,12,14,15,18,20,23,24,26,27,30,32,35,36,38,39,42,44,47,48,50,...}.
(p-4) divides a(p) for prime p = {5,11,23,47,59,71,83,107,131,167,179,191,...} = {5} and all primes from A068231: Primes congruent to 11 (mod 12).
(n+5) divides a(n) for n = {1,2,3,4,5,30,31,45,58,145}.
(n-5) divides a(n) for n = {6,7,9,10,20,25,33,49,57,73,81,97,105,...}.
(p-5) divides a(p) for prime p = {7,73,97,193,241,313,337,409,433,457,577,601,673,769,937,...} = {7} and all primes from A107008: Primes of the form x^2+24y^2. (End)

			Any 2-coloring of the edges of K_6 produces at least two monochromatic triangles.  Having colors induce K_3,3 and 2K_3 shows this is attained, so a(6) = 2.

Alexander Adamchuk, Table of n, a(n) for n = 0..100
R. Ehrenborg, Bounding monochromatic triangles using squares, Math. Magazine, 94 (2021), 383-386.
A. W. Goodman, On Sets of Acquaintances and Strangers at Any Party, Amer. Math. Monthly 66, 778-783, 1959.
L. Sauvé, On chromatic graphs, Amer. Math. Monthly, 68 (1961), 107-111.
A. J. Schwenk, Acquaintance Party Problem, Amer. Math. Monthly 79 (1972), 1113-1117.
V. Vijayalakshmi, Multiplicity of triangles in cocktail party graphs, Discrete Math., 206 (1999), 217-218.
Eric Weisstein's World of Mathematics, Extremal Graph.
Eric Weisstein's World of Mathematics, Monochromatic Forced Triangle.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

Cf. A008804.

Cf. A002144, A141373, A107003, A032766, A007522, A068231, A107008.

[n*(n-1)*(n-2)/6 - Floor((n/2)*Floor(((n-1)/2)^2)): n in [1..20]]; // G. C. Greubel, Oct 06 2017

A049322 := proc(n) local u; if n mod 2 = 0 then u := n/2; RETURN(u*(u-1)*(u-2)/3); elif n mod 4 = 1 then u := (n-1)/4; RETURN(u*(u-1)*(4*u+1)*2/3); else u := (n-3)/4; RETURN(u*(u+1)*(4*u-1)*2/3); fi; end;

Table[Binomial[n,3] - Floor[n/2*Floor[((n-1)/2)^2]],{n,0,100}] (* Alexander Adamchuk, Nov 29 2006 *)

x='x+O('x^99); concat(vector(6), Vec(2*x^6/((x-1)^4*(x+1)^2*(x^2+1)))) \\ Altug Alkan, Apr 08 2016

a(n) = binomial(n,3) - floor(n/2 * floor(((n-1)/2)^2)). - Alexander Adamchuk, Nov 29 2006
G.f.: 2*x^6/((x-1)^4*(x+1)^2*(x^2+1)). - Colin Barker, Nov 28 2012
E.g.f.: ((x - 3)*x^2*cosh(x) - 6*sin(x) + (6 + 3*x - 3*x^2 + x^3)*sinh(x))/24. - Stefano Spezia, May 15 2023

Entry revised by N. J. A. Sloane, Mar 22 2004

A132238 Primes congruent to {11, 13} mod 30.

Original entry on oeis.org

11, 13, 41, 43, 71, 73, 101, 103, 131, 163, 191, 193, 223, 251, 281, 283, 311, 313, 373, 401, 431, 433, 461, 463, 491, 521, 523, 613, 641, 643, 673, 701, 733, 761, 821, 823, 853, 881, 883, 911, 941, 971, 1031, 1033, 1061, 1063

Offset: 1

Omar E. Pol, Aug 15 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A001097, A001359, A006512, A039949, A068231.

[ p: p in PrimesUpTo(1300) | p mod 30 in {11, 13} ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[200]],MemberQ[{11,13}, Mod[#,30]]&]  (* Harvey P. Dale, Mar 12 2011 *)

A138620 Nonnegative integers n such that 12*n-1 is prime.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 9, 11, 14, 15, 16, 19, 20, 21, 22, 26, 29, 30, 32, 35, 36, 37, 39, 40, 41, 42, 47, 49, 50, 54, 55, 57, 60, 62, 69, 70, 72, 74, 76, 79, 81, 82, 85, 86, 91, 92, 96, 97, 99, 102, 105, 107, 109, 110, 114, 119, 120, 121, 124, 125, 126, 127, 130

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 14 2008

nonn

			12*1-1=11, 12*2-1=23, 12*4-1=47, 12*5-1=59, ...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A068231 (resulting primes).

a={};Do[x=12*n-1;If[PrimeQ[x],AppendTo[a,n]],{n,10^2}];a
Select[Range[500],PrimeQ[12#-1]&] (* Harvey P. Dale, May 14 2011 *)

More terms from Harvey P. Dale, May 14 2011

A141798 Numbers n such that 2*23^n + 1 is prime.

Original entry on oeis.org

0, 1, 5, 21, 261, 47589, 93337

Offset: 1

Rick L. Shepherd, Jul 05 2008

Primes found and proved by PrimeForm. No more terms up to 20000.
a(6) and a(7) proved prime by the primality proving program LLR. - Robert Price, Jan 06 2016
a(8) > 2*10^5. - Robert Price, Jan 06 2016

Cf. A141774, A141797, A141802, A190942, A068231.

Join[{0}, Select[Range[1, 261, 2], PrimeQ[2*23^# + 1] &]] (* Arkadiusz Wesolowski, Nov 06 2012 *)

is(n)=ispseudoprime(2*23^n+1) \\ Charles R Greathouse IV, Feb 20 2017

a(6)-a(7) from Robert Price, Jan 06 2016

A138905 a(n) is n-th prime == -1 (mod 6n).

Original entry on oeis.org

5, 23, 71, 167, 179, 431, 461, 863, 863, 839, 1583, 1511, 1949, 2099, 2339, 4127, 4283, 4751, 4673, 4919, 5669, 6599, 8693, 10079, 7349, 10607, 12149, 11087, 12527, 11159, 15809, 19583, 16829, 19583, 13859, 25703, 24197, 25307, 23633, 21839, 34439

Offset: 1

Zak Seidov, Apr 03 2008

nonn

			a(1) = 1st term in A007528 (Primes of form 6n-1)
a(2) = 2nd term in A068231 (Primes congruent to 11 (mod 12))
a(3) = 3rd term in A061242 (Primes of form 18n-1)
a(4) = 4th term in A134517 (Primes of form 24n-1)
a(5) = 5th term in A132236 (Primes congruent to 29 (mod 30))

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A007528, A061242, A068231, A093871, A132236, A134517, A138906, A138907.

a[n_]:=Module[{p=1,cnt=0},Until[cnt==n,If[Mod[Prime[p],6n]==6n-1,cnt++];p++];Prime[p-1]];Array[a,41] (* James C. McMahon, Jun 22 2025 *)

A141774 Numbers n such that 2*11^n + 1 is prime.

Original entry on oeis.org

0, 1, 3, 9, 43, 79, 175, 11325, 13359, 18577

Offset: 1

Rick L. Shepherd, Jul 03 2008

a(11) > 2*10^5. - Robert Price, Oct 16 2015

Cf. A141798, A190942, A068231.

Join[{0}, Select[Range[1, 175, 2], PrimeQ[2*11^# + 1] &]] (* Arkadiusz Wesolowski, Nov 06 2012 *)

for(n=0, 1e5, if(isprime(2*11^n + 1), print1(n", "))) \\ Altug Alkan, Oct 16 2015

a(8)-a(10) from Arkadiusz Wesolowski, Nov 06 2012

A141682 Number of isomorphism classes of (2n+1)-reflexive polygons.

Original entry on oeis.org

16, 1, 12, 29, 1, 61, 81, 1, 113, 131, 2, 163, 50, 2, 215, 233, 2, 34, 285, 3, 317, 335, 2, 367, 182, 3, 419, 72, 4, 469, 489, 3, 93, 539, 4, 571, 591, 3, 185, 641, 5, 673, 131, 5, 725, 240, 6, 148, 795, 5, 827, 845, 3, 877, 897, 7, 929, 186, 6, 338, 656, 7, 240, 1049, 8, 1081, 393, 5, 1133, 1151, 8, 542, 245, 7, 1235, 1253

Offset: 0

Benjamin Nill, Jul 02 2012

nonn

There are no l-reflexive polygons for even index l.

			a(0)=16 equals the number of isomorphism classes of (1-)reflexive polygons, A090045(2).

Andrey Zabolotskiy, Table of n, a(n) for n = 0..99 (from the Graded Ring Database)
Dimitrios I. Dais, On the Twelve-Point Theorem for l-Reflexive Polygons, arXiv:1806.08351 [math.CO], 2018.
A. M. Kasprzyk and B. Nill, Reflexive polytopes of higher index and the number 12, arXiv:1107.4945 [math.AG], 2011.
A. M. Kasprzyk and B. Nill, Toric l-reflexive surfaces at Graded Ring Database.

Cf. A090045.

It seems that for n > 2, a(n) = 17*n - k where k = 21, 22, 23, 24 iff 2*n+1 is a prime from A068228, A068229, A040117, A068231, respectively. - Andrey Zabolotskiy, Apr 21 2022

A160593 Indices of primes congruent to 11 modulo 12.

Original entry on oeis.org

5, 9, 15, 17, 20, 23, 28, 32, 39, 41, 43, 49, 52, 54, 56, 64, 69, 72, 76, 81, 83, 86, 91, 92, 94, 96, 103, 107, 109, 118, 120, 124, 128, 132, 144, 146, 150, 154, 156, 161, 164, 166, 171, 173, 182, 185, 190, 192, 195, 200, 205, 208, 214, 215, 219, 225, 228, 230, 236

Offset: 1

M. F. Hasler, May 22 2009

nonn

The asymptotic density of this sequence is 1/4 (by Dirichlet's theorem). - Amiram Eldar, Mar 02 2021

			a(1) = 5 since the 5th prime, A000040(5) = 11, is the first one to be equal to 11 (mod 12).
a(2) = 9 since the 9th prime, A000040(9) = 23, is the second one to be equal to 11 (mod 12).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A116610 lists the even terms of this sequence, divided by 2.

Cf. A000720, A068231.

Position[Mod[Prime[Range[250]],12],11]//Flatten (* Harvey P. Dale, Apr 13 2022 *)

for( n=1,999, prime(n)%12==11 & print1(n","))

a(n) = A000720(A068231(n)).

cp's OEIS Frontend

A132241 Twin primes congruent to {11, 13} mod 30.

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A167057 Numbers k such that 12*k + 11 is prime.

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A014557 Multiplicity of K_3 in K_n.

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A132238 Primes congruent to {11, 13} mod 30.

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A138620 Nonnegative integers n such that 12*n-1 is prime.

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A141798 Numbers n such that 2*23^n + 1 is prime.

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A138905 a(n) is n-th prime == -1 (mod 6n).

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A141774 Numbers n such that 2*11^n + 1 is prime.

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