OFFSET
1,2
COMMENTS
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.
The b(n) corresponding to each a(n) is A008365(n).
The first 28 entries 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..1920
EXAMPLE
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.
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 always has 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(5) returns this sequence.
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Hans Ruegg, Dec 01 2018
STATUS
approved