A107008 -id:A107008 - cp's OEIS frontend

A014557 Multiplicity of K_3 in K_n.

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

A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

13, 43, 61, 79, 103, 127, 139, 157, 181, 199, 211, 277, 283, 313, 337, 367, 373, 433, 439, 523, 547, 571, 601, 607, 673, 727, 751, 757, 823, 829, 859, 883, 907, 919, 937, 991, 997, 1039, 1063, 1069, 1093, 1117, 1153, 1171, 1213, 1231, 1249, 1291, 1297

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 11; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

193, 337, 457, 673, 1009, 1033, 1129, 1201, 1297, 1801, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2713, 2857, 3049, 3217, 3313, 3361, 3529, 3697, 3889, 4057, 4153, 4201, 4561, 4657, 4729, 4993, 5209, 5233, 5569, 5737, 5881, 6073, 6217, 6337, 6553, 6577

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Also primes of the form x^2 + 168y^2. - T. D. Noe, Apr 29 2008

In base 12, the sequence is 141, 241, 321, 481, 701, 721, 7X1, 841, 901, 1061, 1101, 1201, 12X1, 1461, 1521, 1561, 1681, 16X1, 17X1, 1921, 1X41, 1E01, 1E41, 2061, 2181, 2301, 2421, 24X1, 2521, 2781, 2841, 28X1, 2X81, 3021, 3041, 3281, 33X1, 34X1, 3621, 3721, 3801, 3961, 3981, where X is 10 and E is 11. Moreover, the discriminant is 480. - Walter Kehowski, Jun 01 2008

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

Cf. A139643.

a = {}; w = 26; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a]

The primes are congruent to {1, 25, 121} (mod 168). - T. D. Noe, Apr 29 2008

A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.

Original entry on oeis.org

229, 349, 409, 421, 661, 769, 829, 1021, 1069, 1249, 1381, 1429, 1549, 1789, 1801, 1861, 2089, 2161, 2269, 2389, 3001, 3061, 3109, 3181, 3229, 3469, 3889, 4021, 4129, 4201, 4441, 4861, 4909, 5101, 5449, 5521, 5869, 5881, 6121, 6469, 6481, 6529, 6781

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Are all terms == 1 mod 12? - Zak Seidov, Apr 25 2008

Yes: (i) all terms == 1 mod 3 because the quadratic form has terms == {0,1} mod 3 and the values ==0 mod 3 are not primes. (ii) all terms == 1 mod 4 because the quadratic form has terms == {0,1,2} mod 4 and the values = {0,2} mod 4 are not primes. By the Chinese remainder constructions for coprime 3 and 4 all prime terms are == 1 mod 12. - R. J. Mathar, Jun 10 2020

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references), discriminant 1020.

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 32; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

A139530 Primes of the form 24*k + 13.

Original entry on oeis.org

13, 37, 61, 109, 157, 181, 229, 277, 349, 373, 397, 421, 541, 613, 661, 709, 733, 757, 829, 853, 877, 997, 1021, 1069, 1093, 1117, 1213, 1237, 1381, 1429, 1453, 1549, 1597, 1621, 1669, 1693, 1741, 1789, 1861, 1933, 2029, 2053, 2221, 2269, 2293, 2341, 2389

Offset: 1

Artur Jasinski, Apr 25 2008

nonn

Primes of the form 4x^2+4xy+13y^2. Discriminant=-192. - T. D. Noe, May 02 2008

Also, primes of form u^2+12v^2 with odd v, while A107008 (which is also expressible as x^2+48y^2) has even v. One can transform its form as (2x+y)^2+12y^2 (where y can only be odd), while the second is x^2+12(2y)^2. Both sequences are 1 mod 12 and together they are primes x^2+12y^2 (A068228). [Tito Piezas III, Jan 01 2009]

Robert Price, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139827.

a = {}; Do[If[PrimeQ[24 n + 13], AppendTo[a, 24 n + 13]], {n, 0, 200}]; a
Select[24Range[0,150]+13,PrimeQ]  (* Harvey P. Dale, Mar 11 2011 *)

A289290 Numbers n such that the sum of the divisors of n is of the form m^2+1.

Original entry on oeis.org

1, 657, 3681, 10809, 15777, 17937, 24201, 28521, 54657, 81441, 122697, 154881, 230481, 265257, 336321, 346041, 455337, 473481, 547137, 613017, 718857, 833121, 898137, 1161009, 1226457, 1274841, 1305081, 1510281, 1584801, 1618497, 1695609, 1752417, 1846161, 1965609

Offset: 1

Michel Lagneau, Jul 02 2017

nonn

The corresponding values of m are 0, 31, 73, 125, 151, 161, 187, 203, 281, ...
For n > 1, it appears that a(n) is of the form 9p where p is prime congruent to 1 (mod 24). The corresponding primes p are 73, 409, 1201, 1753, 1993, 2689, 3169, 6073, 9049, 13633, 17209, ... (a subsequence of A107008?).
For n > 1, the divisors of a(n) are of the form {1, 3, 9, p, 3p, 9p} and sigma(a(n)) = 13(1 + p) = 1 + m^2.
Proof:
We use the formula (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 with:
a = 2, b = 3, c = (3m + 2)/13 and d = (2m - 3)/13. We obtain:
a^2 + b^2 = 13, c^2 + d^2 = 1 + p = (m^2 + 1)/13 => p = (m^2 - 12)/13,
(ac - bd)^2 = 1 and (ad + bc)^2 = m^2.
The first terms not of the form 9p are 2493961, 7106353, 8325721, 10708297, and 14120281. Note that solving the Diophantine equation m^2 + 1 = sigma(k)*(1+p), for various numbers k, it is possible to identify other potentially infinite subsequences similar to 9p. For example, 47^2*p, 7^4*p, 3^6*p, 3^2*11^2*p, and so on. - Giovanni Resta, Jul 02 2017

			657 is in the sequence because sigma(657) = 962 = 31^2 + 1.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000203, A002522, A107008.

with(numtheory):nn:=10^6:
for n from 1 to nn do:
    y:=sqrt(sigma(n)-1):
     if y=floor(y) then printf(`%d, `,n):
       else
      fi:
od:

Select[Range[10^6], IntegerQ[Sqrt[DivisorSigma[1, #] - 1]] &] (* Giovanni Resta, Jul 02 2017 *)

isok(n) = issquare(sigma(n) - 1); \\ Michel Marcus, Jul 02 2017

Name edited by Robert Israel, Jul 03 2017

A139505 Primes of the form x^2 + 25x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

151, 163, 307, 397, 409, 541, 547, 601, 673, 811, 823, 859, 967, 997, 1153, 1231, 1237, 1327, 1567, 1669, 1741, 1879, 2083, 2143, 2281, 2293, 2557, 2677, 2707, 2833, 2971, 3037, 3259, 3313, 3433, 3877, 4003, 4129, 4153, 4603, 4639, 4861, 4957, 5101, 5227

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..5000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 25; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)
With[{nn=80},Select[Union[#[[1]]^2+25#[[1]]#[[2]]+#[[2]]^2&/@Tuples[ Range[ 0,nn],2]],PrimeQ[#]&&#Harvey P. Dale, Feb 10 2020 *)

A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

Original entry on oeis.org

7, 5, 7, 17, 13, 29, 19, 41, 73, 31, 97, 191, 43, 89, 97, 109, 61, 311, 137, 73, 149, 241, 337, 181, 197, 103, 313, 109, 331, 229, 257, 397, 139, 281, 151, 457, 317, 821, 337, 349, 181, 547, 193, 389, 199, 401, 1061, 449, 229, 461, 937, 241, 727, 757, 1033, 1321, 271, 1361, 557

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2015

nonn

Least prime q such that q == 1 (mod prime(n) + 1).

Consists of the initial terms of sequences A002476, A002144, A002476, A007519, A068228, A140444, A061237, A141881, A107008, A132230, A133870, A141868, A124826, A142292, A142398, A141948, A088955, A142003, ...

Cf. A263729, A263730, A263769.

Table[q = 2; While[! IntegerQ[(Prime[n]^2 + q Prime@ n)/(Prime@ n + 1)], q = NextPrime@ q]; q, {n, 59}] (* Michael De Vlieger, Oct 26 2015 *)

a(n) = {p = prime(n); q = 2; while ((p^2 + p*q) % (p + 1), q = nextprime(q+1)); q;} \\ Michel Marcus, Oct 26 2015

5 is in this sequence because (prime(2)^2 + 5*prime(2))/(prime(2) + 1) = 6 and 5 is prime.

A139493 Primes of the form x^2 + 9x*y + y^2 for x and y nonnegative.

Original entry on oeis.org

11, 23, 37, 53, 67, 71, 113, 137, 163, 179, 191, 317, 331, 379, 389, 401, 421, 443, 449, 463, 487, 499, 599, 617, 631, 641, 653, 683, 709, 751, 757, 823, 863, 883, 907, 911, 947, 977, 991, 1061, 1087, 1093, 1103, 1171, 1213, 1303, 1367, 1373, 1409, 1423

Offset: 1

Artur Jasinski, Apr 24 2008

nonn

This is a member of the family of sequences of primes of the forms x^2 + kxy + y^2.

See for k=1 A007645 = x^2+3y^2, k=2 squares no primes, k=3 A038872, k=4 A068228 = x^2+9y^2, k=5 A139492, k=6 A007519 = x^2+8y^2, k=7 A033212 = x^2+15y^2, k=8 A107152 = x^2+45y^2, k=9 A139493, k=10 A107008 = x^2+24y^2, k=11 A139494, k=12 A139495, k=13 A139496, k=14* = 10 A107008 = x^2+24y^2, k=15 A139497, k=16 A033215 = x^2+21y^2, k=17 A139498, k=18 A107145 = x^2+40y^2, k=19 A139499, k=20 A139500, k=21 A139501, k=22 A139502, k=23 A139503, k=24 A139504, k=25 A139505, k=26,A139506, k=27 A139507, k=28 A139508, k=29 A139509, k=30 A139510, k=31 A139511, k=32 A139512

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139489, A007645, A068228, A007519, A033212, A033212, A107152, A107008, A033215, A107145, A139490, A139491.

a = {}; w = 9; k = 1; Do[Do[If[PrimeQ[n^2 + w*n*m + k*m^2], AppendTo[a, n^2 + w*n*m + k*m^2]], {n, m, 400}], {m, 1, 400}]; Union[a] (*Artur Jasinski*)

cp's OEIS Frontend

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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A139494 Primes of the form x^2 + 11x*y + y^2 for x and y nonnegative.

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A139506 Primes of the form x^2 + 26x*y + y^2 for x and y nonnegative.

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A139512 Primes of the form x^2 + 32*x*y + y^2 for x and y nonnegative.

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A139530 Primes of the form 24*k + 13.

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A289290 Numbers n such that the sum of the divisors of n is of the form m^2+1.

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Programs

Maple

Mathematica

PARI

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A139505 Primes of the form x^2 + 25x*y + y^2 for x and y nonnegative.

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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).

A139512 Primes of the form x^2 + 32xy + y^2 for x and y nonnegative.