A002476 -id:A002476 - cp's OEIS frontend

A088977 Side of primitive equilateral triangle with prime cevian p=A002476(n) cutting an edge into two integral parts.

8, 15, 21, 35, 40, 48, 65, 77, 80, 91, 112, 117, 119, 133, 160, 168, 171, 187, 207, 209, 221, 224, 253, 255, 264, 280, 312, 323, 325, 341, 352, 377, 391, 403, 408, 425, 435, 440, 455, 465, 483, 504, 525, 527, 560, 576, 595, 609, 624, 645, 651, 665, 667, 703

Offset: 1

Lekraj Beedassy, Oct 31 2003

nonn

The edge a(n) is partitioned into q=s^2 - t^2=A088243(n)*A088296(n) and r=t(2s+t)=A088242(n)*A088299(n) by a cevian of length p. [Alternatively, (p,q,r) form a triangle with angle 2pi/3 opposite side p.] The quadruple {p,q,r,a(n)=q+r} satisfies the triangle relation: see A061281, or the simpler relation a(n)^2 = p^2 + q*r.

Jean-François Alcover, Table of n, a(n) for n = 1..2918
F. Barnes, Deriving 60 degree triples

Cf. A002476, A088241, A088242, A088243, A088296, A088298, A088299.

sol[p_] := Solve[0 < t < s && s^2 + s t + t^2 == p, {s, t}, Integers];
Union[Reap[For[n = 1, n <= 10000, n++, If[PrimeQ[p = 6n + 1], an = s(s + 2t) /. sol[p][[1]]]; Sow[an]]][[2, 1]]] (* Jean-François Alcover, Mar 06 2020 *)

a(n) = A088241(n)*A088298(n) = s(s+2t), where s^2 + st + t^2, with s>t, form the primes p = 1 (mod 6) = A002476(n).

More terms from Ray Chandler, Nov 01 2003

A088241 Values of y, where x^2 + xy + y^2 = p (xA002476).

Original entry on oeis.org

2, 3, 3, 5, 4, 6, 5, 7, 8, 7, 8, 9, 7, 7, 10, 9, 12, 11, 11, 9, 13, 14, 11, 12, 15, 10, 12, 13, 17, 16, 11, 13, 17, 13, 17, 15, 12, 15, 20, 13, 18, 17, 21, 21, 18, 17, 21, 14, 21, 19, 24, 23, 19, 22, 15, 18, 20, 21, 19, 25, 18, 19, 23, 21, 27, 17, 27, 25, 19, 20, 27, 23, 28, 21, 26

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

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

Cf. A002476, A088242, A088243, A088296, A088298, A088299, A088977.

R:= NULL: count:= 0:
for k from 1 while count < 100 do
  p:= 6*k+1;
  if not isprime(p) then next fi;
  S:= select(t -> subs(t,x) > 0 and subs(t,x) < subs(t,y), [isolve(x^2+x*y+y^2=p)]);
  S:= map(t -> subs(t,y), S);
  R:= R,op(S); count:= count+1;
od:
R; #Robert Israel, Jun 16 2025

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers]; Sow[y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088242 Values of x, where x^2 + xy + y^2 = p (xA002476).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 1, 3, 3, 2, 5, 6, 3, 5, 1, 3, 4, 7, 2, 1, 6, 5, 1, 9, 7, 6, 1, 3, 10, 8, 3, 9, 4, 7, 11, 8, 1, 11, 5, 7, 1, 2, 7, 9, 4, 13, 5, 8, 1, 3, 9, 5, 14, 11, 9, 8, 11, 3, 13, 12, 7, 10, 1, 15, 2, 6, 14, 13, 4, 10, 3, 13, 7, 17, 3, 7, 9, 13, 8, 11, 16, 15, 6, 3, 12, 17, 7, 9, 1, 3, 16

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

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

Cf. A002476, A088241, A088243, A088296, A088298, A088299, A088977.

R:= NULL: count:= 0:
for k from 1 while count < 100 do
  p:= 6*k+1;
  if not isprime(p) then next fi;
  S:= select(t -> subs(t,x) > 0 and subs(t,x) < subs(t,y), [isolve(x^2+x*y+y^2=p)]);
  S:= map(t -> subs(t,x), S);
   R:= R,op(S); count:= count+1;
od:
R; # Robert Israel, Jun 16 2025

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers]; Sow[x /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088243 Values of x + y, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

3, 4, 5, 6, 7, 7, 9, 9, 9, 10, 11, 11, 12, 13, 13, 14, 13, 14, 15, 16, 15, 15, 17, 17, 16, 19, 19, 19, 18, 19, 21, 21, 20, 22, 21, 22, 23, 23, 21, 24, 23, 24, 22, 23, 25, 26, 25, 27, 26, 27, 25, 26, 28, 27, 29, 29, 29, 29, 30, 28, 31, 31, 30, 31, 28, 32, 29, 31, 33, 33, 31, 33

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088296, A088298, A088299, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers]; Sow[x + y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088296 Values of y - x, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

1, 2, 1, 4, 1, 5, 1, 5, 7, 4, 5, 7, 2, 1, 7, 4, 11, 8, 7, 2, 11, 13, 5, 7, 14, 1, 5, 7, 16, 13, 1, 5, 14, 4, 13, 8, 1, 7, 19, 2, 13, 10, 20, 19, 11, 8, 17, 1, 16, 11, 23, 20, 10, 17, 1, 7, 11, 13, 8, 22, 5, 7, 16, 11, 26, 2, 25, 19, 5, 7, 23, 13, 25, 8, 19, 1, 26, 20, 17, 10, 19, 14, 5, 7, 23

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088243, A088298, A088299, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers];
Sow[y - x /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 09 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088298 Values of 2x + y, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

4, 5, 7, 7, 10, 8, 13, 11, 10, 13, 14, 13, 17, 19, 16, 19, 14, 17, 19, 23, 17, 16, 23, 22, 17, 28, 26, 25, 19, 22, 31, 29, 23, 31, 25, 29, 34, 31, 22, 35, 28, 31, 23, 25, 32, 35, 29, 40, 31, 35, 26, 29, 37, 32, 43, 40, 38, 37, 41, 31, 44, 43, 37, 41, 29, 47, 31, 37, 47, 46, 35

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088243, A088296, A088299, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers];
Sow[2x + y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 09 2020 *)

More terms from Ray Chandler, Nov 04 2003

A088299 Values of x + 2y, where x^2 + xy + y^2=p (xA002476).

Original entry on oeis.org

5, 7, 8, 11, 11, 13, 14, 16, 17, 17, 19, 20, 19, 20, 23, 23, 25, 25, 26, 25, 28, 29, 28, 29, 31, 29, 31, 32, 35, 35, 32, 34, 37, 35, 38, 37, 35, 38, 41, 37, 41, 41, 43, 44, 43, 43, 46, 41, 47, 46, 49, 49, 47, 49, 44, 47, 49, 50, 49, 53, 49, 50, 53, 52, 55, 49, 56, 56, 52, 53

Offset: 1

Lekraj Beedassy, Nov 03 2003

nonn

Cf. A002476, A088241, A088242, A088243, A088296, A088298, A088977.

Reap[For[n = 1, n <= 200, n++, If[PrimeQ[p = 6 n + 1], s = Solve[x^2 + x y + y^2 == p && 0 < x < y, {x, y}, Integers];
Sow[x + 2y /. s[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 09 2020 *)

More terms from Ray Chandler, Nov 04 2003

A108164 Semiprimes p*q where both p and q are primes of the form 6n+1 (A002476).

Original entry on oeis.org

49, 91, 133, 169, 217, 247, 259, 301, 361, 403, 427, 469, 481, 511, 553, 559, 589, 679, 703, 721, 763, 793, 817, 871, 889, 949, 961, 973, 1027, 1057, 1099, 1141, 1147, 1159, 1261, 1267, 1273, 1333, 1339, 1351, 1369, 1387, 1393, 1417, 1477, 1501, 1561, 1591

Offset: 1

Jonathan Vos Post, Jun 13 2005

These are the products of terms from A107890 excluding multiples of 3.
Every semiprime not divisible by 2 or 3 must be in one of these three disjoint sets:
A108164 = the product of two primes of the form 6n+1 (A002476),
A108166 = the product of two primes of the form 6n-1 (A007528),
A108172 = the product of a prime of the form 6n+1 and a prime of the form 6n-1.
The product of two primes of the form 6n+1 is a semiprime of the form 6n+1.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

Robert Israel, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.

Cf. A001358, A002476, A107890.

N:= 2000: # To get all terms <= N
P:= select(isprime, [seq(i,i=7..N/7, 6)]):
sort(select(`<=`,[seq(seq(P[i]*P[j],j=1..i),i=1..nops(P))],N)); # Robert Israel, Dec 27 2018

With[{nn=50},Take[Times@@@Tuples[Select[6*Range[nn]+1,PrimeQ],2]// Union,nn]] (* Harvey P. Dale, May 20 2021 *)

{a(n)} = {p*q where both p and q are in A002476}.

Edited and extended by Ray Chandler, Oct 15 2005

A334477 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).

Original entry on oeis.org

1, 0, 0, 3, 6, 0, 2, 5, 4, 0, 2, 2, 1, 2, 5, 9, 8, 9, 6, 7, 0, 4, 3, 2, 3, 9, 3, 3, 3, 3, 2, 1, 8, 7, 8, 5, 9, 1, 7, 0, 5, 3, 9, 4, 7, 7, 1, 1, 7, 5, 0, 8, 7, 2, 1, 3, 7, 0, 2, 2, 4, 0, 2, 6, 4, 1, 6, 5, 2, 3, 7, 1, 7, 3, 7, 1, 7, 3, 6, 2, 6, 1, 4, 6, 6, 2, 7, 5, 2, 0, 4, 0, 8, 1, 5, 1, 4, 8, 2, 9, 8, 9, 1, 5, 7

Offset: 1

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A002476(k)^(2*s+1)) / (1 - 1/A002476(k)^(2*s+1)) = sqrt(3) * (2*Pi)^(2*s + 1) * zeta(2*s + 1) * A002114(s) / ((2^(2*s + 1) + 1) * (3^(2*s + 1) + 1) * (2*s)! * zeta(4*s + 2)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) / (1 - 1/A002476(k)^s) = (zeta(s, 1/6) - zeta(s, 5/6))*zeta(s) / ((2^s + 1)*(3^s + 1)*zeta(2*s)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) * (1 + 1/A007528(k)^s) = 6^s * zeta(s) / ((2^s + 1) * (3^s + 1) * zeta(2*s)).
For s > 0, Product_{k>=1} ((A007528(k)^(2*s+1) - 1) / (A007528(k)^(2*s+1) + 1)) * ((A002476(k)^(2*s+1) + 1) / (A002476(k)^(2*s+1) - 1)) = 6 * A002114(s)^2 * (4*s + 2)! / ((2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2) = Bernoulli(2*s)^2 * (4*s + 2)! * (zeta(2*s + 1, 1/6) - zeta(2*s + 1, 5/6))^2 / (8*Pi^2 * (2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2 * zeta(2*s)^2).

			1.0036025402212598967043239333321878591705394771...

Cf. A002476, A175646, A334424, A334426, A334478, A334481.

A334477 / A334478 = 15*sqrt(3)*zeta(3)/Pi^3.

A334477 * A334479 = 810*zeta(3)/Pi^6.

More digits from Vaclav Kotesovec, Jun 27 2020

A334478 Decimal expansion of Product_{k>=1} (1 - 1/A002476(k)^3).

Original entry on oeis.org

9, 9, 6, 4, 0, 1, 6, 9, 2, 8, 1, 6, 0, 3, 6, 6, 3, 2, 2, 6, 2, 3, 6, 1, 1, 2, 2, 3, 8, 4, 7, 1, 8, 7, 9, 9, 9, 6, 5, 5, 7, 3, 8, 1, 8, 7, 1, 4, 0, 5, 3, 1, 5, 3, 7, 8, 6, 9, 8, 8, 9, 7, 4, 9, 3, 0, 1, 5, 9, 1, 3, 3, 2, 5, 3, 4, 3, 0, 6, 8, 4, 2, 5, 6, 2, 1, 9, 1, 9, 7, 2, 9, 9, 7, 7, 5, 2, 3, 2, 2, 1, 2, 3, 0, 1, 9

Offset: 0

Vaclav Kotesovec, May 02 2020

In general, for s > 0, Product_{k>=1} (1 + 1/A002476(k)^(2*s+1)) / (1 - 1/A002476(k)^(2*s+1)) = sqrt(3) * (2*Pi)^(2*s + 1) * zeta(2*s + 1) * A002114(s) / ((2^(2*s + 1) + 1) * (3^(2*s + 1) + 1) * (2*s)! * zeta(4*s + 2)).
For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) / (1 - 1/A002476(k)^s) = (zeta(s, 1/6) - zeta(s, 5/6))*zeta(s) / ((2^s + 1)*(3^s + 1)*zeta(2*s)).
For s > 1, Product_{k>=1} (1 - 1/A002476(k)^s) * (1 - 1/A007528(k)^s) = 6^s / ((2^s - 1)*(3^s - 1)*zeta(s)).

			0.996401692816036632262361122384718799965573818714...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26 (case 3 1 3 = 1/A334478).

Cf. A002476, A175646, A334425, A334427, A334477.

A334477 / A334478 = 15*sqrt(3)*zeta(3)/Pi^3.

A334478 * A334480 = 108/(91*zeta(3)).

More digits from Vaclav Kotesovec, Jun 27 2020

cp's OEIS Frontend

A088977 Side of primitive equilateral triangle with prime cevian p=A002476(n) cutting an edge into two integral parts.

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A088241 Values of y, where x^2 + xy + y^2 = p (xA002476).

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A088242 Values of x, where x^2 + xy + y^2 = p (xA002476).

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A088243 Values of x + y, where x^2 + xy + y^2=p (xA002476).

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A088296 Values of y - x, where x^2 + xy + y^2=p (xA002476).

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A088298 Values of 2x + y, where x^2 + xy + y^2=p (xA002476).

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A088299 Values of x + 2y, where x^2 + xy + y^2=p (xA002476).

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A108164 Semiprimes p*q where both p and q are primes of the form 6n+1 (A002476).

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A334477 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).

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