A322275 - OEIS

%I #24 Sep 11 2022 12:04:44

%S 1,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,

%T 107,109,113,127,131,137,139,149,151,157,67,167,173,179,181,191,193,

%U 197,199,211,223,83,229,233,239,241,251,257,263,269,271,277,281,283,1,293,307,311,313,317,683,331,337,107,349,353,239,1,103,277,331,47,389

%N Smallest multiplication factors f, prime or 1, for all b (mod 120120), coprime to 120120 (= 4*13#), so that b*f is a square mod 8, and modulo all primes up to 13.

%C See sequence A322269 for further explanations. This sequence is related to A322269(6).

%C The sequence is periodic, repeating itself after phi(120120) terms. Its largest term is 3583, which is A322269(6). In order to satisfy the conditions, both f and b must be coprime to 120120. Otherwise, the product would be zero mod a prime <= 13.

%C The b(n) corresponding to each a(n) is A008366(n).

%C The first 32 terms are trivial: f=b, and then the product b*f naturally is a square modulo everything.

%H Hans Ruegg, <a href="/A322275/b322275.txt">Table of n, a(n) for n = 1..23040</a>

%e The 44th number coprime to 120120 is 227. a(44) is 83, because 83 is the smallest prime by which we can multiply 227, so that the product (227*83 = 18841) is a square mod 8, and modulo all primes up to 13.

%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(6) returns this sequence.

%o \\ Similarly, sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(5) returns A322274.

%Y Cf. A322269, A322271, A322272, A322273, A322274, A008366.

%K nonn,fini,full

%O 1,2

%A _Hans Ruegg_, Dec 01 2018