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

1,2

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.

Links

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

Example

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.

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(6) returns this sequence.
\\ Similarly, sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(5) returns A322274.

Crossrefs

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

Sequence in context: A054484 A054796 A352379 * A008366 A126769 A092216

Adjacent sequences: A322272 A322273 A322274 * A322276 A322277 A322278

Keyword

nonn,fini,full

Author

Hans Ruegg, Dec 01 2018

Status

approved