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

1,2

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.

Links

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

Example

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.

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

Crossrefs

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

Sequence in context: A084374 A063193 A056758 * A096489 A008364 A140461

Adjacent sequences: A322270 A322271 A322272 * A322274 A322275 A322276

Keyword

nonn,fini,full

Author

Hans Ruegg, Dec 01 2018

Status

approved