A322274 - OEIS

%I #24 Sep 11 2022 12:05:08

%S 1,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,

%T 103,107,109,113,127,131,113,19,29,79,157,67,167,1,173,179,181,71,193,

%U 197,31,211,389,103,83,181,233,239,241,463,59,257,263,269,271,277,281,283,1,173,131,283,311,97,53,443,331,193,107,61,257,239,1,103,277

%N Smallest multiplication factors f, prime or 1, for all b (mod 9240), coprime to 9240 (= 4*11#), so that b*f is a square mod 8, mod 3, mod 5, mod 7, and mod 11.

%C See sequence A322269 for further explanations. This sequence is related to A322269(5).

%C The sequence is periodic, repeating itself after phi(9240) terms. Its largest term is 1873, which is A322269(5). In order to satisfy the conditions, both f and b must be coprime to 9240. Otherwise, the product would be zero mod a prime <= 11.

%C The b(n) corresponding to each a(n) is A008365(n).

%C The first 28 entries are trivial: f=b, and then the product b*f naturally is a square modulo everything.

%H Hans Ruegg, <a href="/A322274/b322274.txt">Table of n, a(n) for n = 1..1920</a>

%e The 30th number coprime to 9240 is 139. a(30) is 19, because 19 is the smallest prime by which we can multiply 139, so that the product (139*19 = 2641) is a square mod 8, and modulo all primes up to 11.

%o (PARI)

%o QresCode(n, nPrimes) = {

%o   code = bitand(n,7)>>1;

%o   for (j=2, nPrimes,

%o     x = Mod(n,prime(j));

%o     if (issquare(x), code += (1<<j));

%o   );

%o   return (code);

%o }

%o QCodeArray(n) = {

%o   totalEntries = 1<<(n+1);

%o   f = vector(totalEntries);

%o   f[totalEntries-3] = 1;  \\ 1 always has the same code: ...111100

%o   counter = 1;

%o   forprime(p=prime(n+1), +oo,

%o     code = QresCode(p, n);

%o     if (f[code+1]==0,

%o       f[code+1]=p;

%o       counter += 1;

%o       if (counter==totalEntries, return(f));

%o     )

%o   )

%o }

%o sequence(n) = {

%o   f = QCodeArray(n);

%o   primorial = prod(i=1, n, prime(i));

%o   entries = eulerphi(4*primorial);

%o   a = vector(entries);

%o   i = 1;

%o   forstep (x=1, 4*primorial-1, 2,

%o     if (gcd(x,primorial)==1,

%o       a[i] = f[QresCode(x, n)+1];

%o       i += 1;

%o     );

%o   );

%o   return(a);

%o }

%o \\ sequence(5) returns this sequence.

%o \\ Similarly, sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.

%Y Cf. A322269, A322271, A322272, A322273, A322275, A008365.

%K nonn,fini,full

%O 1,2

%A _Hans Ruegg_, Dec 01 2018