A322271 Smallest multiplication factors f, prime or 1, for all b (mod 24), coprime to 24, so that b*f is a nonzero square mod 8 and mod 3.

1, 5, 7, 11, 13, 17, 19, 23

Offset

1,2

Comments

See sequence A322269 for further explanations. This sequence is related to A322269(2).

The sequence is periodic, repeating itself after phi(24) terms. Its largest term is 23, which is A322269(2). In order to satisfy the conditions, both f and b must be coprime to 24.

The b(n) corresponding to each a(n) is A007310(n).

In this case, the sequence is trivial, since each term is being multiplied by itself. The next related sequence, A322272, corresponding to A322269(3), already has several nontrivial terms.

Links

Table of n, a(n) for n=1..8.

Example

The 4th number coprime to 24 is 11. a(4) is 11, because 11 is the smallest prime by which we can multiply 11, so that the product (11*11 = 121) is a square mod 8 and mod 3.

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(2) returns A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.

Crossrefs

Cf. A322269, A322272, A322273, A322274, A322275, A007310.

Sequence in context: A320048 A246351 A272260 * A306289 A136801 A106571

Adjacent sequences: A322268 A322269 A322270 * A322272 A322273 A322274

Keyword

nonn,fini,full

Author

Hans Ruegg, Dec 01 2018

Status

approved