A141373 -id:A141373 - cp's OEIS frontend

A107154 Duplicate of A141373.

3, 19, 43, 67, 139, 163, 211, 283, 307, 331, 379, 499, 523, 547, 571, 619, 643, 691, 739, 787, 811, 859, 883, 907, 1051, 1123, 1171, 1291, 1459, 1483, 1531, 1579, 1627, 1699, 1723, 1747, 1867, 1987, 2011, 2083, 2131, 2179, 2203, 2251, 2347, 2371

Offset: 1

dead

A107132 Primes of the form 2x^2 + 13y^2.

Original entry on oeis.org

2, 13, 31, 149, 167, 317, 359, 397, 463, 487, 509, 613, 661, 709, 839, 1061, 1087, 1103, 1151, 1181, 1367, 1471, 1783, 1789, 1861, 2039, 2111, 2221, 2269, 2437, 2503, 2621, 2647, 2917, 2927, 2957, 3023, 3079, 3167, 3229, 3373, 3541, 3853

Offset: 1

T. D. Noe, May 13 2005

Discriminant = -104. Binary quadratic forms ax^2+cy^2 have discriminant d=-4ac. We consider sequences of primes produced by forms with -400<=d<=0, a<=c and gcd(a,c)=1. These restrictions yield 173 sequences of prime numbers, which are organized by discriminant below. See A106856 for primes of the form ax^2+bxy+cy^2 with discriminant > -100.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.
L. E. Dickson, History of the Theory of Numbers, Vol. 3, Chelsea, 1923.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A033218 (d=-104), A014752 (d=-108), A107133, A107134 (d=-112), A033219 (d=-116), A107135-A107137, A033220 (d=-120), A033221 (d=-124), A105389 (d=-128), A107138, A033222 (d=-132), A107139, A033223 (d=-136), A107140, A033224 (d=-140), A107141, A107142 (d=-144), A033225 (d=-148), A107143, A033226 (d=-152), A033227 (d=-156), A107144, A107145 (d=-160), A033228 (d=-164), A107146-A107148, A033229 (d=-168).
Cf. A033230 (d=-172), A107149, A107150 (d=-176), A107151, A107152 (d=-180), A107153, A033231 (d=-184), A033232 (d=-188), A141373 (d=-192), A107155 (d=-196), A107156, A107157 (d=-200), A107158, A033233 (d=-204), A107159, A107160 (d=-208), A033234 (d=-212), A107161, A107162 (d=-216), A033235 (d=-220), A107163, A107164 (d=-224), A107165, A033236 (d=-228), A107166, A033237 (d=-232), A033238 (d=-236).
Cf. A107167-A107169 (d=-240), A033239 (d=-244), A107170, A033240 (d=-248), A014754 (d=-256), A107171, A033241 (d=-260), A107172-A107174, A033242 (d=-264), A033243 (d=-268), A107175, A107176 (d=-272), A107177, A033244 (d=-276), A107178-A107180, A033245 (d=-280), A033246 (d=-284), A107181 (d=-288), A033247 (d=-292), A107182, A033248 (d=-296), A107183, A107184 (d=-300), A107185, A107186 (d=-304), A107187, A033249 (d=-308).
Cf. A107188-A107190, A033250 (d=-312), A033251 (d=-316), A107191, A107192 (d=-320), A107193 (d=-324), A107194, A033252 (d=-328), A033253 (d=-332), A107195-A107198 (d=-336), A107199, A033254 (d=-340), A107200, A033255 (d=-344), A033256 (d=-348), A107132 A107201, A107202 (d=-352), A033257 (d=-356), A107203-A107206 (d=-360), A107207, A033258 (d=-364), A107208, A107209 (d=-368), A107210, A033202 (d=-372).
Cf. A107211, A033204 (d=-376), A033206 (d=-380), A107212, A107213 (d=-384), A033208 (d=-388), A107214, A107215 (d=-392), A107216, A107217 (d=-396), A107218, A107219 (d=-400).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

QuadPrimes2[2, 0, 13, 10000] (* see A106856 *)

list(lim)=my(v=List([2,13]),t); for(y=1,sqrtint(lim\13), for(x=1,sqrtint((lim-13*y^2)\2), if(isprime(t=2*x^2+13*y^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

A140633 Primes of the form 7x^2+4xy+52y^2.

Original entry on oeis.org

7, 103, 127, 223, 367, 463, 487, 607, 727, 823, 967, 1063, 1087, 1303, 1327, 1423, 1447, 1543, 1567, 1663, 1783, 2143, 2287, 2383, 2503, 2647, 2767, 2887, 3343, 3463, 3583, 3607, 3727, 3823, 3847, 3943, 3967, 4327, 4423, 4447, 4567, 4663

Offset: 1

T. D. Noe, May 19 2008

Discriminant=-1440. Also primes of the forms 7x^2+6xy+87y^2 and 7x^2+2xy+103y^2.
Voight proves that there are exactly 69 equivalence classes of positive definite binary quadratic forms that represent almost the same primes. 48 of those quadratic forms are of the idoneal type discussed in A139827. The remaining 21 begin at A140613 and end here. The cross-references section lists the quadratic forms in the same order as tables 1-6 in Voight's paper. Note that A107169 and A139831 are in the same equivalence class.
In base 12, the sequence is 7, 87, X7, 167, 267, 327, 347, 427, 507, 587, 687, 747, 767, 907, 927, 9X7, X07, X87, XX7, E67, 1047, 12X7, 13X7, 1467, 1547, 1647, 1727, 1807, 1E27, 2007, 20X7, 2107, 21X7, 2267, 2287, 2347, 2367, 2607, 2687, 26X7, 2787, 2847, where X is for 10 and E is for 11. Moreover, the discriminant is X00 and that all elements are {7, 87, X7, 167, 187, 247} mod 260. - Walter Kehowski, May 31 2008

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
John Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

Cf. A033205, A007519, A068228, A107135, A107144, A107152, A107151.
Cf. A107181, A139502, A139856, A139854, A139874, A139877, A139897.
Cf. A140613, A140614. A140615, A139923, A140616-A140619, A139988.
Cf. A139993, A140008, A140010, A140620-A140632, A033212, A102273.
Cf. A107007, A107003, A139830, A107169, A139831, A141373, A139847.
Cf. A139850, A139855, A139860, A139879, A139880, A139924, A139990.
Cf. A139998, A139992, A139996, A140003, A140013, A107006, A139858.
Cf. A107008, A140633, A139991, A139857, A139859, A139878, A007645, A002313.

Union[QuadPrimes2[7, 4, 52, 10000], QuadPrimes2[7, -4, 52, 10000]] (* see A106856 *)

A107003 Primes of the form 24n + 5.

Original entry on oeis.org

5, 29, 53, 101, 149, 173, 197, 269, 293, 317, 389, 461, 509, 557, 653, 677, 701, 773, 797, 821, 941, 1013, 1061, 1109, 1181, 1229, 1277, 1301, 1373, 1493, 1613, 1637, 1709, 1733, 1877, 1901, 1949, 1973, 1997, 2069, 2141, 2213, 2237, 2309, 2333, 2357, 2381, 2477

Offset: 1

T. D. Noe, May 09 2005

Primes of the form 5x^2+2xy+5y^2, with x and y any integer. Discriminant=-96. Also primes of the forms 5x^2+4xy+20y^2 and 5x^2+2xy+29y^2. See A140633. - T. D. Noe, May 19 2008

Also primes of the form -4*x^2+4*x*y+5*y^2, of discriminant -96 (as well as of the form 8*x^2+16*x*y+5*y^2). - Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

			29 is a member because we can write 29=-4*4^2+4*4*3+5*3^2 (or 29=8*1^2+16*1*1+5*1^2).

Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.

Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 [First 1000 terms from Vincenzo Librandi]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A141373, A141375, A141376 (d = -96).

Union[QuadPrimes2[5, 2, 5, 10000], QuadPrimes2[5, -2, 5, 10000]] (* see A106856 *)
Select[24*Range[0,200]+5,PrimeQ] (* Harvey P. Dale, Aug 25 2025 *)

select(n->n%24==5, primes(1000)) \\ Charles R Greathouse IV, Dec 07 2014

a(n) ~ 8n log n. - Charles R Greathouse IV, Dec 07 2014

Name and comment switched by Charles R Greathouse IV, Dec 07 2014

Edited by N. J. A. Sloane, Jul 14 2019

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

A141375 Primes of the form x^2 + 8xy - 8y^2 (as well as of the form x^2 + 10x*y + y^2).

Original entry on oeis.org

73, 97, 193, 241, 313, 337, 409, 433, 457, 577, 601, 673, 769, 937, 1009, 1033, 1129, 1153, 1201, 1249, 1297, 1321, 1489, 1609, 1657, 1753, 1777, 1801, 1873, 1993, 2017, 2089, 2113, 2137, 2161, 2281, 2377, 2473, 2521, 2593, 2617, 2689, 2713, 2833, 2857

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

Conjecture: Same as A107008. - Arkadiusz Wesolowski, Jul 25 2012
Discriminant = +96.
x^2 + 8*x*y - 8*y^2 = (x+4*y)^2 - 24*y^2, and x^2 + 10*x*y + y^2 = (x+5*y)^2 - 24*y^2, so this sequence is also primes of the form x^2 - 24*y^2. - Michael Somos, Jun 05 2013

			a(1) = 73 because we can write 73 = 5^2 + 8*5*2 - 8*2^2 (or 73 = 2^2 + 10*2*3 + 3^2).

Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107008, A141373, A107003, A141376 (d = -96).

Union[Select[Flatten[Table[x^2 + 8*x*y - 8*y^2, {x, 40}, {y, 40}]], # > 0 && PrimeQ[#] &]] (* T. D. Noe, Jun 12 2013 *)

More terms and offset corrected by Arkadiusz Wesolowski, Jul 25 2012

A141376 Primes of the form -x^2 + 8xy + 8y^2 (as well as of the form 15x^2 + 24xy + 8*y^2).

Original entry on oeis.org

23, 47, 71, 167, 191, 239, 263, 311, 359, 383, 431, 479, 503, 599, 647, 719, 743, 839, 863, 887, 911, 983, 1031, 1103, 1151, 1223, 1319, 1367, 1439, 1487, 1511, 1559, 1583, 1607, 1823, 1847, 1871, 2039, 2063, 2087, 2111, 2207, 2351, 2399, 2423, 2447, 2543

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

Discriminant = +96.
Values of the quadratic form are {0, 8, 12, 15, 20, 23} mod 24, so this is a subsequence of A134517. - R. J. Mathar, Jul 30 2008
Is this the same sequence as A134517?
Substituting 2y = y' gives the quadratic form A141171, so these terms are a subsequence of the terms in A141171. - R. J. Mathar, Jun 10 2020

			a(2)=47 because we can write 47 = -1^2 + 8*1*2 + 8*2^2 (or 47 = 15*1^2 + 24*1*1 + 8*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107003, A134517, A141171, A141373, A141375 (d = -96).

More terms from Arkadiusz Wesolowski, Jul 25 2012

A139527 Numbers n such that numbers 24n+5 are primes.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 11, 12, 13, 16, 19, 21, 23, 27, 28, 29, 32, 33, 34, 39, 42, 44, 46, 49, 51, 53, 54, 57, 62, 67, 68, 71, 72, 78, 79, 81, 82, 83, 86, 89, 92, 93, 96, 97, 98, 99, 103, 106, 109, 112, 114, 116, 118, 119, 121, 123, 134, 141, 142, 144, 147, 148, 149, 153, 154

Offset: 1

Artur Jasinski, Apr 25 2008

nonn

Numbers n such that:
24n+1 is prime see A111174, primes 24n+1 see A107008
24n+5 is prime see A139527, primes 24n+5 see A107003
24n+7 is prime see A139483, primes 24n+7 see A107006
24n+11 is prime A139528, primes 24n+11 see A107007
24n+13 is prime see A139529, primes 24n+13 see A139530
24n+17 is prime see A139531, primes 24n+17 see A107181
24n+19 is prime see A139532, primes 24n+19 see A141373
24n+23 is prime see A131210, primes 24n+23 see A134517

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

a = {}; Do[If[PrimeQ[24 n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Table[(Prime[n]-5)/24,{n,800}],IntegerQ] (* Harvey P. Dale, Feb 25 2016 *)

cp's OEIS Frontend

A107154 Duplicate of A141373.

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A107132 Primes of the form 2x^2 + 13y^2.

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A140633 Primes of the form 7x^2+4xy+52y^2.

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A107003 Primes of the form 24n + 5.

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A014557 Multiplicity of K_3 in K_n.

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A141375 Primes of the form x^2 + 8*x*y - 8*y^2 (as well as of the form x^2 + 10*x*y + y^2).

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A141376 Primes of the form -x^2 + 8*x*y + 8*y^2 (as well as of the form 15*x^2 + 24*x*y + 8*y^2).

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A139527 Numbers n such that numbers 24n+5 are primes.

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A141375 Primes of the form x^2 + 8xy - 8y^2 (as well as of the form x^2 + 10x*y + y^2).

A141376 Primes of the form -x^2 + 8xy + 8y^2 (as well as of the form 15x^2 + 24xy + 8*y^2).