

A322274


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.


7



1, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 113, 19, 29, 79, 157, 67, 167, 1, 173, 179, 181, 71, 193, 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
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OFFSET

1,2


COMMENTS

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



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[totalEntries3] = 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*primorial1, 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



STATUS

approved



