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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 (list; graph; refs; listen; history; text; internal format)
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[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.
\\ Similarly, sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.
CROSSREFS
Sequence in context: A272119 A075761 A046064 * A008365 A132077 A235154
KEYWORD
nonn,fini,full
AUTHOR
Hans Ruegg, Dec 01 2018
STATUS
approved

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Last modified April 22 06:39 EDT 2024. Contains 371888 sequences. (Running on oeis4.)