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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. 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 (list; graph; refs; listen; history; text; internal format)
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 with 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 has always 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

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Last modified June 19 15:55 EDT 2021. Contains 345143 sequences. (Running on oeis4.)