A007522 -id:A007522 - 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

A115257 Partial sums of binomial(2n,n)^2.

Original entry on oeis.org

1, 5, 41, 441, 5341, 68845, 922621, 12701245, 178338145, 2542242545, 36677022081, 534311328705, 7846771001041, 116019251361041, 1725360846921041, 25786805857871441, 387084441100423541, 5832802431123111941

Offset: 0

Paul Barry, Jan 18 2006

Central coefficients of number triangle A115255.
p divides all a(n) from a((p-1)/2) to a(p-1) for Gaussian primes p=7,23,31,79,167,431,479,983, ... of the form 4n+3, A002145(n) and for primes of the form 8n+7, A007522(n). - Alexander Adamchuk, Jul 05 2006
Conjecture: For any positive integer n, the polynomials Sum_{k=0}^n binomial(2k,k)^2*x^k and Sum_{k=0}^n binomial(2k,k)^2*x^k/(k+1) are irreducible over the field of rational numbers. - Zhi-Wei Sun, Mar 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital Library of Mathematical Functions, Elliptic Integrals, NIST, 2016.

Cf. A000984, A002145, A007522, A115255, A228002.

series( 2*EllipticK(4*x^(1/2))/(Pi*(1-x)) ,x=0,20); # Mark van Hoeij, Apr 06 2013

Table[Sum[((2k)!/(k!)^2)^2,{k,0,n}], {n,0,40}] (* Alexander Adamchuk, Jul 05 2006 *)
Accumulate[(Binomial[2#,#])^2&/@Range[0,20]]  (* Harvey P. Dale, Mar 04 2011 *)

makelist(sum(binomial(2*k,k)^2,k,0,n),n,0,12); /* Emanuele Munarini, Oct 28 2016 */

a(n) = sum(k=0, n, binomial(2*k, k)^2); \\ Michel Marcus, Oct 30 2016

a(n) = Sum_{k=0..n} C(2k, k)^2. a(n) = A115255(2n, n).
a(n) = C(2n,n)^2 + C(2n-2,n-1)^2 + ... + C(2k,k)^2 + ... + C(2,1)^2 + C(0,0)^2, where C(2k,k) = (2k)!/(k!)^2 are the central binomial coefficients A000984(k). - Alexander Adamchuk, Jul 05 2006
a(n) = Sum_{k=0..n} ((2k)!/(k!)^2)^2. a(n) = Sum_{k=0..n} A000984[k]^2. - Alexander Adamchuk, Jul 05 2006
Recurrence: n^2*a(n) = (17*n^2-16*n+4)*a(n-1) - 4*(2*n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 16^(n+1)/(15*Pi*n). - Vaclav Kotesovec, Oct 19 2012
From Emanuele Munarini, Oct 28 2016: (Start)
Let K(x) be the complete elliptic integral of the first kind as defined in [DLMF, 19.2.4] for phi = Pi/2.
a(n) = (2/Pi)*K(16)-((16^(n+1)*Gamma(n+3/2)^2)/(Pi*Gamma(n+2)^2))*hypergeometric (1,n+3/2,n+3/2;n+2,n+2;16).
G.f.: A(t) = (2/Pi)*(K(16*t)/(1-t)).
Diff. eq. satisfied by the g.f. t*(1-17*t+16*t^2)*A''(t)+(1-35*t+64*t^2)*A'(t)-(5-36*t)*A(t)=0. (End)

A167860 Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).

Original entry on oeis.org

7, 47, 191, 383, 439, 1151, 1399, 2351, 2879, 3119, 3511, 3559, 4127, 5087, 5431, 6911, 8887, 9127, 9791, 9887, 12391, 13151, 14407, 15551, 16607, 19543, 20399, 21031, 21319, 21839, 23039, 25391, 26399, 28087, 28463, 28711, 29287, 33223, 39551, 43103, 44879, 46271

Offset: 1

Alexander Adamchuk, Nov 13 2009

nonn

Apparently A167860 is a subset of primes of the form 8*k + 7 (A007522).
Every A167859(m) from m=(p-1)/2 to m=(p-1) is divisible by prime p belonging to A167860.
7^3 divides A167859(13) and 7^2 divides A167859(10)-A167859(13).
Every A167859(m) from m=(kp-1 - (p-1)/2) to m=(kp-1) is divisible by prime p from A167860.
Every A167859(m) from m=((p^2-1)/2) to m=(p^2-1) is divisible by prime p from A167860. For p=7 every A167859(m) from m=((p^3-1)/2) to m=(p^3-1) and from m=((p^4-1)/2) to m(p^4-1)is divisible by p^2.

R. J. Mathar, Table of n, a(n) for n = 1..57

Cf. A000984, A006134, A007522, A066796, A082590, A132310, A144635, A167713, A167859, A276649.

A167859 := proc(n)
    option remember;
    if n <= 1 then
        add( (binomial(2*k, k)/2^k)^2, k=0..n) ;
        4^n*% ;
    else
        4*(5*n^2 - 4*n + 1)*procname(n-1) - 16*(2*n - 1)^2*procname(n-2) ;
        %/n^2 ;
    end if;
end proc:
isA167860 := proc(p)
    local m ;
    for m from (p-1)/2 to p-1 do
        if modp(A167859(m),p) > 0 then
            return false;
        end if;
    end do:
    true ;
end proc:
A167860 := proc(n)
    option remember ;
    if n = 0 then
        2;
    else
        p := nextprime(procname(n-1)) ;
        while not isA167860(p) do
            p := nextprime(p) ;
        end do ;
        return p;
    end if;
end proc:
seq(A167860(n),n=1..10) ; # R. J. Mathar, Jan 22 2025

is(p) = if(isprime(p)&&p%2, my(m=Mod(1, p), s=m); for(k=1, p\2, s+=(m*=(2*k-1)/k)^2); !s, 0); \\ Jinyuan Wang, Jul 24 2022

More terms from Jinyuan Wang, Jul 24 2022

A269454 Safe primes that are not congruent to -1 mod 8.

Original entry on oeis.org

5, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, 2027, 2099, 2459, 2579, 2819, 2963, 3203, 3467, 3779, 3803, 3947, 4139, 4259, 4283, 4547, 4787, 5099, 5387, 5483, 5507, 5939, 6659, 6779, 6827, 6899, 7187, 7523

Offset: 1

Marina Ibrishimova, Feb 27 2016

nonn

For safe primes see A005385.
Conjecture: If p and q are two distinct safe primes not congruent to -1 mod 8 then the order of 2 mod p*q is phi(p*q)/2. For phi see A000010.
Note: The order of 2 mod p*q is the smallest positive integer k such that 2^k = 1 mod p*q. See Rosen's definition of the order of an integer on p.334. Also, k is smaller than or equal to phi(p*q)/2 for all products of distinct odd primes p and q. See Cohen's Prop. 1.4.2 on p. 25.
2^(phi(p*q)/2) == 1 (mod p*q) for all distinct odd primes p and q. See Nagell's corollary to Theorem 64, p. 106, with a = 2 and n = p*q. - Wolfdieter Lang, Mar 31 2016

Henri Cohen, Graduate Texts In Mathematics: A Course in Computational Algebraic Number Theory, Springer, 2000, p. 25
Trygve Nagell, Introduction to Number Theory, Chelsea, 1964, p. 106.
Kenneth H. Rosen, Elementary Number Theory And Its Applications, AT&T Laboratories, 2005, p. 334

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005385, A007522.

[ p: p in PrimesUpTo(8000) | IsPrime((p-1) div 2) and not p mod 8 eq 7]; // Vincenzo Librandi, Feb 28 2016

Select[Prime@ Range@ 1000, And[PrimeQ[(# - 1)/2], MemberQ[Range[0, 6], Mod[#, 8]]] &] (* Michael De Vlieger, Feb 28 2016 *)

lista(nn) = {forprime(p=3, nn, if (((p % 8) != 7) && isprime((p-1)/2), print1(p, ", ")););} \\ Michel Marcus, Mar 24 2016

A005385 without its intersection with A007522.

More terms from Vincenzo Librandi, Feb 28 2016

A343104 Smallest number having exactly n divisors of the form 8*k + 1.

Original entry on oeis.org

1, 9, 81, 153, 891, 1377, 8019, 3825, 11025, 15147, 88209, 31977, 354375, 99225, 121275, 95931, 7144929, 187425, 893025, 287793, 1403325, 1499553, 1715175, 675675, 1091475, 6024375, 1576575, 1686825, 72335025, 2027025, 2264802453041139, 2297295, 11609325, 121463793, 9823275

Offset: 1

Jianing Song, Apr 05 2021

nonn

Smallest index of n in A188169.
a(n) exists for all n, since 3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 3^0, 3^2, ..., 3^(2n-2). This actually gives an upper bound for a(n).
From David A. Corneth, Apr 05 2021: (Start)
All terms are odd since if a term is even then the odd part has the same number of such divisors.
No a(2*k + 1) is divisible by a prime congruent to 1 (mod 8).
If for some k, A188169(k) > m then A188169(k*t) > m for all t > 0. This can be used to trim searches when looking for some a(m).
If gcd(k, m) = 1 then A188169(k) * A188169(m) <= A188169(k*m) (End)

			a(4) = 153 since it is the smallest number with exactly 4 divisors congruent to 1 modulo 8, namely 1, 9, 17 and 153.

Smallest number having exactly n divisors of the form 8*k + i: this sequence (i=1), A343105 (i=3), A343106 (i=5), A188226 (i=7).
Cf. A188169.
Cf. A007519, A007520, A007521, A007522.

res(n,a,b) = sumdiv(n, d, (d%a) == b)
a(n) = if(n>0, for(k=1, oo, if(res(k,8,1)==n, return(k))))

a(2n-1) <= 3^(2n-2) * 11, since 3^(2n-2) * 11 has exactly 2n-1 divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^(2n-2), 3^1 * 11, 3^3 * 11, ..., 3^(2n-3) * 11.

a(2n) <= 3^(n-1) * 187, since 3^(n-1) * 187 has exactly 2n divisors congruent to 1 modulo 8: 3^0, 3^2, ..., 3^b, 3^0 * 17, 3^2 * 17, ..., 3^b * 17, 3^1 * 11, 3^3 * 11, ..., 3^a * 11, 3^1 * 187, 3^3 * 187, ... 3^a * 187, where a is the largest odd number <= n-1 and b is the largest even number <= n-1.

More terms from David A. Corneth, Apr 06 2021

A065907 First solution mod p of x^4 = 2 for primes p such that only two solutions exist.

Original entry on oeis.org

2, 8, 15, 17, 15, 3, 48, 4, 16, 34, 33, 47, 98, 92, 68, 63, 114, 78, 153, 157, 107, 36, 156, 115, 86, 58, 222, 297, 57, 6, 18, 235, 66, 142, 221, 395, 227, 33, 120, 408, 368, 131, 301, 408, 253, 149, 318, 405, 459, 121, 30, 206, 122, 28, 543, 472, 88, 283, 696, 246

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three times in this sequence. Confirmed for the first 2399 terms of A007522 (primes < 100000). There are integers which do occur thrice, e.g. 221, 1159.

			a(8) = 4, since 127 is the eighth term of A007522 and 4 is the first solution mod 127 of x^4 = 2.

Cf. A040098, A007522, A065908.

a065907(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2] == 2,print1(s[1],",")))
a065907(1600)

a(n) = first (least) solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has only two solutions mod p, i.e. p is the n-th term of A007522.

A065908 Second solution mod p of x^4 = 2 for primes p such that only two solutions exist.

Original entry on oeis.org

5, 15, 16, 30, 56, 76, 55, 123, 135, 133, 158, 152, 125, 147, 195, 208, 197, 281, 214, 226, 324, 403, 307, 364, 401, 445, 377, 310, 574, 641, 701, 492, 677, 609, 602, 444, 636, 854, 791, 511, 599, 852, 690, 623, 786, 914, 769, 698, 692, 1102, 1201, 1073

Offset: 1

Klaus Brockhaus, Nov 29 2001

nonn

Conjecture: no integer occurs more than three times in this sequence. Confirmed for the first 2399 terms of A007522 (primes < 100000). In this section, there are no integers which do occur thrice.

			a(3) = 16, since 31 is the third term of A007522 and 16 is the second solution mod 31 of x^4 = 2.

A040098, A007522, A065907.

a065908(m) = local(s); forprime(p = 2,m,s = []; for(x = 0,p-1, if(x^4%p == 2%p,s = concat(s,[x]))); if(matsize(s)[2] == 2,print1(s[2],",")))
a065908(1400)

a(n) = second (largest) solution mod p of x^4 = 2, where p is the n-th prime such that x^4 = 2 has only two solutions mod p, i.e. p is the n-th term of A007522.

A127582 a(n) = the smallest prime number of the form k*2^n - 1, for k >= 1.

Original entry on oeis.org

2, 3, 3, 7, 31, 31, 127, 127, 1279, 3583, 5119, 6143, 8191, 8191, 81919, 131071, 131071, 131071, 524287, 524287, 14680063, 14680063, 109051903, 109051903, 654311423, 738197503, 738197503, 2147483647, 2147483647, 2147483647

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

			a(0)=2 because 2 = 3*2^0 - 1 is prime.
a(1)=3 because 3 = 2*2^1 - 1 is prime.
a(2)=3 because 3 = 1*2^2 - 1 is prime.
a(3)=7 because 7 = 1*2^3 - 1 is prime.
a(4)=31 because 31 = 2*2^4 - 1 is prime.

Robert Israel, Table of n, a(n) for n = 0..3310

Cf. A007522, A127575-A127581, A127583-A127587.

A087522 is identical except for a(1).

p:= 2: A[0]:= 2:
for n from 1 to 100 do
  if p+1 mod 2^n = 0 then A[n]:= p
  else
    p:=p+2^(n-1);
    while not isprime(p) do p:= p+2^n od:
    A[n]:= p;
  fi
od:
seq(A[i],i=0..100); # Robert Israel, Jan 13 2017

a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k 2^n + 2^n - 1], {n, 0, 50}]; a (* Artur Jasinski, Jan 19 2007 *)

a(n) << 37^n by Xylouris's improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

Edited by Don Reble, Jun 11 2007

Further edited by N. J. A. Sloane, Jul 03 2008

A254930 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1 (primes congruent to 1 or 7 mod 8).

Original entry on oeis.org

5, 7, 11, 9, 13, 17, 13, 19, 23, 17, 15, 21, 25, 17, 23, 27, 35, 23, 29, 21, 41, 25, 31, 23, 35, 29, 39, 43, 37, 31, 27, 49, 53, 33, 31, 37, 47, 41, 55, 59, 31, 45, 39, 49, 37, 35, 61, 37, 35

Offset: 1

Wolfdieter Lang, Feb 12 2015

The corresponding terms y = y2(n) are given in A254931(n).
There is only one fundamental solution for prime 2 (no second class exists), and this solution (x, y) has been included in (A002334(1), A002335(1)) = (2, 1).
The second class x sequence for the primes 1 (mod 8), which are given in A007519, is A254762, and for the primes 7 (mod 8), given in A007522, it is A254766.
The second class solutions give the second smallest positive integer solutions of this Pell equation.
For comments and the Nagell reference see A254760.

			n = 3: 11^2 - 2*7^2 = 23 = A001132(3) = A007522(2).
The first pairs of these second class solutions [x2(n), y2(n)] are (a star indicates primes congruent to 1 (mod 8)):
n  A001132(n)   a(n)  A254931(n)
1     7           5        3
2    17 *         7        4
3    23          11        7
4    31           9        5
5    41 *        13        8
6    47          17       11
7    71          13        7
8    73 *        19       12
9    89 *        17       10
10   97 *        15        8
11  103          21       13
12  113 *        25       16
13  127          17        9
14  137 *        23       14
15  151          27       17
16  167          35       23
17  191          23       13
18  193 *        29       18
19  199          21       11
20  223          41       27
...

Cf. A001132, A254931, A002334, A002335, A007519, A254762, A007522, A254766, A254760.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; x2 = xy[[-1, 1]] // Simplify; Print[x2]; Sow[x2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

a(n)^2 - 2*A254931(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer solving this (generalized) Pell equation.

a(n) = 3*A002334(n+1) - 4*A002335(n+1), n >= 1.

A254931 Fundamental positive solution y = y2(n) of the second class of the Pell equation x^2 - 2*y^2 = A001132(n), n >= 1, (primes congruent to 1 or 7 mod 8).

Original entry on oeis.org

3, 4, 7, 5, 8, 11, 7, 12, 15, 10, 8, 13, 16, 9, 14, 17, 23, 13, 18, 11, 27, 14, 19, 12, 22, 17, 25, 28, 23, 18, 14, 32, 35, 19, 17, 22, 30, 25, 36, 39, 16, 28, 23, 31, 21, 19, 40, 20, 18, 38

Offset: 1

Wolfdieter Lang, Feb 12 2015

The corresponding terms x = x2(n) are given in A254930(n).
The y2-sequence for the second class for the primes congruent to 1 (mod 8), which are given in A007519, is 2*A254763. For the primes congruent to 7 (mod 8), given in A007522, the y2-sequence is A254929.
For comments and the Nagell reference see A254760.

			a(4) = 2*7 - 3*3 = 5.
A254930(4)^2 - 2*a(4)^2 = 9^2 - 2*5^2 = 31 = A001132(4) = A007522(3).
See A254930 for the first pairs (x2(n), y2(n)).

Cf. A254930, A001132, A002334, A002335, A007519, 2*A254763, A007522, A254929, A254760.

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[MatchQ[Mod[p, 8], 1|7], rp = Reduce[x > 0 && y > 0 && x^2 - 2 y^2 == p, {x, y}, Integers]; If[rp =!= False, xy = {x, y} /. {ToRules[rp /. C[1] -> 1]}; y2 = xy[[-1, 2]] // Simplify; Print[y2]; Sow[y2]]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2019 *)

A254930(n)^2 - 2*a(n)^2 = A001132(n), and a(n) is the second largest (proper) positive integer satisfying this (generalized) Pell equation.

a(n) = 2*A002334(n+1) - 3*A002335(n+1), n >= 1.

cp's OEIS Frontend

A014557 Multiplicity of K_3 in K_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A115257 Partial sums of binomial(2n,n)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A167860 Primes p dividing every A167859(m) from m=(p-1)/2 to m=(p-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Extensions

A269454 Safe primes that are not congruent to -1 mod 8.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A343104 Smallest number having exactly n divisors of the form 8*k + 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A065907 First solution mod p of x^4 = 2 for primes p such that only two solutions exist.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A065908 Second solution mod p of x^4 = 2 for primes p such that only two solutions exist.

Views

Author

Keywords

Comments