cp's OEIS Frontend

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.

%I A322271 #26 Sep 11 2022 12:06:49
%S A322271 1,5,7,11,13,17,19,23
%N 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.
%C A322271 See sequence A322269 for further explanations. This sequence is related to A322269(2).
%C A322271 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.
%C A322271 The b(n) corresponding to each a(n) is A007310(n).
%C A322271 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.
%e A322271 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.
%o A322271 (PARI)
%o A322271 QresCode(n, nPrimes) = {
%o A322271   code = bitand(n,7)>>1;
%o A322271   for (j=2, nPrimes,
%o A322271     x = Mod(n,prime(j));
%o A322271     if (issquare(x), code += (1<<j));
%o A322271   );
%o A322271   return (code);
%o A322271 }
%o A322271 QCodeArray(n) = {
%o A322271   totalEntries = 1<<(n+1);
%o A322271   f = vector(totalEntries);
%o A322271   f[totalEntries-3] = 1;  \\ 1 always has the same code: ...111100
%o A322271   counter = 1;
%o A322271   forprime(p=prime(n+1), +oo,
%o A322271     code = QresCode(p, n);
%o A322271     if (f[code+1]==0,
%o A322271       f[code+1]=p;
%o A322271       counter += 1;
%o A322271       if (counter==totalEntries, return(f));
%o A322271     )
%o A322271   )
%o A322271 }
%o A322271 sequence(n) = {
%o A322271   f = QCodeArray(n);
%o A322271   primorial = prod(i=1, n, prime(i));
%o A322271   entries = eulerphi(4*primorial);
%o A322271   a = vector(entries);
%o A322271   i = 1;
%o A322271   forstep (x=1, 4*primorial-1, 2,
%o A322271     if (gcd(x,primorial)==1,
%o A322271       a[i] = f[QresCode(x, n)+1];
%o A322271       i += 1;
%o A322271     );
%o A322271   );
%o A322271   return(a);
%o A322271 }
%o A322271 \\ sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.
%Y A322271 Cf. A322269, A322272, A322273, A322274, A322275, A007310.
%K A322271 nonn,fini,full
%O A322271 1,2
%A A322271 _Hans Ruegg_, Dec 01 2018