A320162 Irregular triangle read by rows: row n lists 0 <= k < p^2 such that p^2 divides A172236(k, p-Kronecker(k^2+4, p)), p = prime(n).

0, 0, 4, 5, 0, 7, 18, 0, 12, 20, 24, 25, 29, 37, 0, 5, 18, 19, 24, 43, 78, 97, 102, 103, 116, 0, 2, 14, 45, 70, 82, 87, 99, 124, 155, 167, 0, 24, 38, 40, 63, 83, 103, 105, 184, 186, 206, 226, 249, 251, 265, 0, 31, 37, 63, 79, 100, 137, 144, 150, 180, 181, 211, 217, 224, 261, 282, 298, 324, 330

Offset

1,3

Comments

p always divides A172236(k, p-Kronecker(k^2+4, p)), so it's interesting to see when p^2 also divides A172236(k, p-Kronecker(k^2+4, p)). If this holds, then p is called a k-Wall-Sun-Sun prime (and thus being a (k + s*p^2)-Wall-Sun-Sun prime for all integer s). Specially, there is no Wall-Sun-Sun prime below 10^14 implies that there is no 1 in the first pi(10^14) rows.

Note that A172236(0, m) is not defined, so here k must be understood as a remainder modulo p^2. because A172236(k+s*p^2, m) == A172236(k, m) (mod p^2).

Let p = prime(n). Every row contains 0. For n >= 2, if p == 3 (mod 4), then the n-th row contains p numbers, whose remainders modulo p form a permutation of {0, 1, 2, 3, ..., p - 2, p - 1}. If p == 1 (mod 4), then the n-th row contains p - 2 numbers, whose remainders modulo p form a permutation of {0, 1, 2, 3, ..., p - 2, p - 1} \ {+-2*((p - 1)/2)! mod p}.

Every row is antisymmetric, that is, k is a member iff p^2 - k is, k > 0. As a result, the sum of the n-th row is prime(n)^2*(prime(n) - 1)/2 if prime(n) == 3 (mod 4) and prime(n)^2*(prime(n) - 3)/2 if prime(n) == 1 (mod 4).

Equivalently, if p = prime(n) == 1 (mod 4), then row n lists 0 <= k < p^2 such that p^2 divides A172236(k, (p-Kronecker(k^2+4, p))/2). - Jianing Song, Jul 06 2019

Links

Jianing Song, Table of n, a(n) for n = 1..29193 (primes below 600)

Jianing Song, Table of n, a(n) for n = 1..75966 (primes below 1000)

Wikipedia, Wall-Sun-Sun prime

Example

Table starts
p = 2: 0,
p = 3: 0, 4, 5,
p = 5: 0, 7, 18,
p = 7: 0, 12, 20, 24, 25, 29, 37,
p = 11: 0, 5, 18, 19, 24, 43, 78, 97, 102, 103, 116,
p = 13: 0, 2, 14, 45, 70, 82, 87, 99, 124, 155, 167,
p = 17: 0, 24, 38, 40, 63, 83, 103, 105, 184, 186, 206, 226, 249, 251, 265,
p = 19: 0, 31, 37, 63, 79, 100, 137, 144, 150, 180, 181, 211, 217, 224, 261, 282, 298, 324, 330,
p = 23: 0, 21, 30, 38, 40, 70, 79, 89, 111, 149, 198, 248, 281, 331, 380, 418, 440, 450, 459, 489, 491, 499, 508,
p = 29: 0, 15, 40, 41, 49, 51, 56, 64, 74, 84, 126, 182, 204, 381, 460, 637, 659, 715, 757, 767, 777, 785, 790, 792, 800, 801, 826,
...

Prog

(PARI)
B(k, p) = (([k, 1; 1, 0]^(p-kronecker(k^2+4, p)))[1, 2])%(p^2)
forprime(p=2, 50, for(k=0, p^2-1, if(!B(k, p), print1(k, ", "))); print)

Crossrefs

Cf. A143548, A172236, A320161 (discriminant k^2-4).

Cf. A238736 (primes p such that 2 occurs in the corresponding row).

Sequence in context: A240160 A249860 A360962 * A354068 A200013 A178219

Adjacent sequences: A320159 A320160 A320161 * A320163 A320164 A320165

Keyword

nonn,tabf

Author

Jianing Song, Oct 06 2018

Status

approved