A322275 Smallest multiplication factors f, prime or 1, for all b (mod 120120), coprime to 120120 (= 4*13#), so that b*f is a square mod 8, and modulo all primes up to 13.

1, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 67, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 83, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 1, 293, 307, 311, 313, 317, 683, 331, 337, 107, 349, 353, 239, 1, 103, 277, 331, 47, 389

Offset: 1

Author

Hans Ruegg, Dec 01 2018

Keywords

• nonn
• fini
• full

Comments

See sequence A322269 for further explanations. This sequence is related to A322269(6).

The sequence is periodic, repeating itself after phi(120120) terms. Its largest term is 3583, which is A322269(6). In order to satisfy the conditions, both f and b must be coprime to 120120. Otherwise, the product would be zero mod a prime <= 13.

The b(n) corresponding to each a(n) is A008366(n).

The first 32 terms are trivial: f=b, and then the product b*f naturally is a square modulo everything.

Examples

			The 44th number coprime to 120120 is 227. a(44) is 83, because 83 is the smallest prime by which we can multiply 227, so that the product (227*83 = 18841) is a square mod 8, and modulo all primes up to 13.
		

Links

• Hans Ruegg, Table of n, a(n) for n = 1..23040

Crossrefs

Cf. A322269, A322271, A322272, A322273, A322274, A008366.

Programs

PARI

QresCode(n, nPrimes) = {
  code = bitand(n,7)>>1;
  for (j=2, nPrimes,
    x = Mod(n,prime(j));
    if (issquare(x), code += (1<A322271, sequence(3) returns A322272, ... sequence(5) returns A322274.