A371497 Irregular triangle read by rows: n-th row gives congruence classes s such that the n-th prime q is a quadratic residue modulo an odd prime p if and only if p = plus or minus s for some s (mod m), where m = q if q is of the form 4k + 1, else m = 4q.
1, 1, 1, 1, 3, 9, 1, 5, 7, 9, 19, 1, 3, 4, 1, 2, 4, 8, 1, 3, 5, 9, 15, 17, 25, 27, 31, 1, 7, 9, 11, 13, 15, 19, 25, 29, 41, 43, 1, 4, 5, 6, 7, 9, 13, 1, 3, 5, 9, 11, 15, 23, 25, 27, 33, 41, 43, 45, 49, 55, 1, 3, 4, 7, 9, 10, 11, 12, 16, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 1, 3, 7, 9, 13, 17
Offset: 1
Examples
The 1st prime, 2, not of the form 4k + 1, is a square modulo odd primes p if and only if p = +/- 1 (mod 4*2 = 8). The 6th prime, 13, of the form 4k + 1, is a square modulo odd primes p if and only if p = +/- 1, +/- 3, or +/- 4 (mod 13). The irregular triangle T(n,k) begins (q is prime(n)): n q \k 1 2 3 4 5 6 7 8 9 10 11 1, 2: 1 2, 3: 1 3, 5: 1 4, 7: 1 3 9 5, 11: 1 5 7 9 19 6: 13: 1 3 4 7, 17: 1 2 4 8 8, 19: 1 3 5 9 15 17 25 27 31 9, 23: 1 7 9 11 13 15 19 25 29 41 43 10, 29: 1 4 5 6 7 9 13
Programs
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Python
from sympy import prime def A371497_row(n): q = prime(n) res = {i*i % q for i in range(1, q//2 + 1)} if q % 4 == 1: res = {a for a in res if 2*a < q} else: res = {((a % 4 - 1) * q + a) % (4*q) for a in res} res = {a if a < 2*q else 4*q - a for a in res} return sorted(res)
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