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 A322272 Smallest multiplication factors f, prime or 1, for all a (mod 120), coprime to 120, so that b*f is a nonzero square mod 8, mod 3, and mod 5. 6

%I #26 Sep 11 2022 12:05:30

%S 1,7,11,13,17,19,23,29,31,13,41,43,23,1,53,11,61,43,71,73,53,31,83,41,

%T 19,73,29,7,83,61,17,71

%N Smallest multiplication factors f, prime or 1, for all a (mod 120), coprime to 120, so that b*f is a nonzero square mod 8, mod 3, and mod 5.

%C See sequence A322269 for further explanations. This sequence is related to A322269(3).

%C The sequence is periodic, repeating itself after phi(120) terms. Its largest term is 83, which is A322269(3). In order to satisfy the conditions, both f and b must be coprime to 120. Otherwise, the product would be zero mod a prime <= 5.

%C The b(n) corresponding to each a(n) is A007775(n).

%e The 10th number coprime to 120 is 37. a(10) is 13, because 13 is the smallest prime by which we can multiply 37, so that the product (37*13 = 481) is a square mod 8, mod 3 and mod 5.

%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(3)

%Y Cf. A322269, A322271, A322273, A322274, A322275, A007775.

%K nonn,fini,full

%O 1,2

%A _Hans Ruegg_, Dec 01 2018

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