A322274 Smallest multiplication factors f, prime or 1, for all b (mod 9240), coprime to 9240 (= 4*11#), so that b*f is a square mod 8, mod 3, mod 5, mod 7, and mod 11.

1, 13, 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, 113, 19, 29, 79, 157, 67, 167, 1, 173, 179, 181, 71, 193, 197, 31, 211, 389, 103, 83, 181, 233, 239, 241, 463, 59, 257, 263, 269, 271, 277, 281, 283, 1, 173, 131, 283, 311, 97, 53, 443, 331, 193, 107, 61, 257, 239, 1, 103, 277

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(5).

The sequence is periodic, repeating itself after phi(9240) terms. Its largest term is 1873, which is A322269(5). In order to satisfy the conditions, both f and b must be coprime to 9240. Otherwise, the product would be zero mod a prime <= 11.

The b(n) corresponding to each a(n) is A008365(n).

The first 28 entries are trivial: f=b, and then the product b*f naturally is a square modulo everything.

Examples

			The 30th number coprime to 9240 is 139. a(30) is 19, because 19 is the smallest prime by which we can multiply 139, so that the product (139*19 = 2641) is a square mod 8, and modulo all primes up to 11.
		

Links

• Hans Ruegg, Table of n, a(n) for n = 1..1920

Crossrefs

Cf. A322269, A322271, A322272, A322273, A322275, A008365.

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(6) returns A322275.