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

1,2

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.

Links

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

Example

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.

Prog

(PARI)
QresCode(n, nPrimes) = {
  code = bitand(n, 7)>>1;
  for (j=2, nPrimes,
    x = Mod(n, prime(j));
    if (issquare(x), code += (1<<j));
  );
  return (code);
}
QCodeArray(n) = {
  totalEntries = 1<<(n+1);
  f = vector(totalEntries);
  f[totalEntries-3] = 1;  \\ 1 always has the same code: ...111100
  counter = 1;
  forprime(p=prime(n+1), +oo,
    code = QresCode(p, n);
    if (f[code+1]==0,
      f[code+1]=p;
      counter += 1;
      if (counter==totalEntries, return(f));
    )
  )
}
sequence(n) = {
  f = QCodeArray(n);
  primorial = prod(i=1, n, prime(i));
  entries = eulerphi(4*primorial);
  a = vector(entries);
  i = 1;
  forstep (x=1, 4*primorial-1, 2,
    if (gcd(x, primorial)==1,
      a[i] = f[QresCode(x, n)+1];
      i += 1;
    );
  );
  return(a);
}
\\ sequence(5) returns this sequence.
\\ Similarly, sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.

Crossrefs

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

Sequence in context: A272119 A075761 A046064 * A008365 A132077 A235154

Adjacent sequences: A322271 A322272 A322273 * A322275 A322276 A322277

Keyword

nonn,fini,full

Author

Hans Ruegg, Dec 01 2018

Status

approved