A322272 Smallest multiplication factors f, prime or 1, for all a (mod 120), coprime to 120, so that b*f is a nonzero square mod 8, mod 3, and mod 5.

1, 7, 11, 13, 17, 19, 23, 29, 31, 13, 41, 43, 23, 1, 53, 11, 61, 43, 71, 73, 53, 31, 83, 41, 19, 73, 29, 7, 83, 61, 17, 71

Offset

1,2

Comments

See sequence A322269 for further explanations. This sequence is related to A322269(3).

The sequence is periodic, repeating itself after phi(120) terms. Its largest term is 83, which is A322269(3). In order to satisfy the conditions, both f and b must be coprime to 120. Otherwise, the product would be zero mod a prime <= 5.

The b(n) corresponding to each a(n) is A007775(n).

Links

Table of n, a(n) for n=1..32.

Example

The 10th number coprime to 120 is 37. a(10) is 13, because 13 is the smallest prime by which we can multiply 37, so that the product (37*13 = 481) is a square mod 8, mod 3 and mod 5.

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(3)

Crossrefs

Cf. A322269, A322271, A322273, A322274, A322275, A007775.

Sequence in context: A128974 A286609 A005776 * A161850 A007775 A070884

Adjacent sequences: A322269 A322270 A322271 * A322273 A322274 A322275

Keyword

nonn,fini,full

Author

Hans Ruegg, Dec 01 2018

Status

approved