A242090 -id:A242090 - cp's OEIS frontend

A030006 a(n) = (prime(n)-1)*(prime(n)-5)/12.

0, 1, 5, 8, 16, 21, 33, 56, 65, 96, 120, 133, 161, 208, 261, 280, 341, 385, 408, 481, 533, 616, 736, 800, 833, 901, 936, 1008, 1281, 1365, 1496, 1541, 1776, 1825, 1976, 2133, 2241, 2408, 2581, 2640, 2945, 3008, 3136, 3201, 3605, 4033, 4181, 4256, 4408, 4641

Offset: 3

N. J. A. Sloane

nonn

Confirmed a(n) = A242090(n) for 3 <= n <= 4000. - Fausto A. C. Cariboni, Feb 23 2019

T. D. Noe, Table of n, a(n) for n = 3..1000

Cf. A000040.

Table[(p - 1) (p - 5)/12, {p, Prime[Range[3, 50]]}] (* T. D. Noe, Apr 16 2012 *)
((#-1)(#-5))/12&/@Prime[Range[3,60]] (* Harvey P. Dale, Aug 13 2023 *)

A242089 Number of triples (a,b,c) with 0 < a < b < c < p and a + b + c == 0 mod p, where p = prime(n).

Original entry on oeis.org

0, 0, 0, 2, 10, 16, 32, 42, 66, 112, 130, 192, 240, 266, 322, 416, 522, 560, 682, 770, 816, 962, 1066, 1232, 1472, 1600, 1666, 1802, 1872, 2016, 2562, 2730, 2992, 3082, 3552, 3650, 3952, 4266, 4482, 4816, 5162, 5280, 5890, 6016, 6272, 6402, 7210, 8066, 8362, 8512

Offset: 1

Jonathan Sondow, Jun 16 2014

a(n) is even. (Proof. Each triple (a,b,c) with b < p/2 pairs uniquely with a triple (a',b',c') = (p-c,p-b,p-a) with b' > p/2.)

			For prime(4) = 7 there are 2 triples (a,b,c) with 0 < a < b < c < 7 and a + b + c == 0 mod 7, namely, 1+2+4 = 7 and 3+5+6 = 2*7, so a(4) = 2.

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..1000
Steven J. Miller, Combinatorial and Additive Number Theory Problem Sessions, arXiv:1406.3558 [math.NT], 2014-2023. See Nathan Kaplan's Problem 2014.1.4 on p. 30.

Cf. A069905, A242090.

Table[ Length[ Reduce[ Mod[a + b + c, Prime[n]] == 0 && 0 < a < b < c < Prime[n], {a, b, c}, Integers]], {n, 40}]

a(n) = 2 * round((prime(n) - 3)^2/12) \\ David A. Corneth, May 27 2025

a(n) = 2*A242090(n).

a(n) = 2*A069905(prime(n)-3) = 2 * round((prime(n) - 3)^2/12). - David A. Corneth, May 27 2025

a(41)-a(50) from Fausto A. C. Cariboni, Sep 30 2018

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A030006 a(n) = (prime(n)-1)*(prime(n)-5)/12.

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A242089 Number of triples (a,b,c) with 0 < a < b < c < p and a + b + c == 0 mod p, where p = prime(n).

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