A322273 Smallest multiplication factors f, prime or 1, for all b (mod 840), coprime to 840 (= 4*7#), so that b*f is a nonzero square mod 8, mod 3, mod 5, and mod 7.

1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 43, 71, 73, 79, 83, 41, 73, 101, 103, 107, 109, 113, 1, 127, 59, 113, 19, 47, 29, 79, 13, 43, 47, 1, 173, 11, 61, 283, 71, 193, 53, 31, 41, 211, 29, 103, 83, 61, 113, 71, 241, 127, 59, 37, 17, 23

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(4).

The sequence is periodic, repeating itself after phi(840) = 192 terms. Its largest term is 311, which is A322269(4). In order to satisfy the conditions, both f and b must be coprime to 840. Otherwise, the product would be zero mod a prime <= 7.

The b(n) corresponding to each a(n) is A008364(n).

The first 15 terms are trivial: f=b, and then the product b*f naturally is a square modulo everything.

Examples

			The 16th number coprime to 840 is 67. a(16) is 43, because 43 is the smallest prime by which we can multiply 67, so that the product (67*43 = 2881) is a square mod 8, mod 2, mod 3, mod 5, and mod 7.
		

Links

• Hans Ruegg, Table of n, a(n) for n = 1..192

Crossrefs

Cf. A322269, A322271, A322272, A322274, A322275, A008364.

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.