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A322275 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. 7
1, 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, 137, 139, 149, 151, 157, 67, 167, 173, 179, 181, 191, 193, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
See sequence A322269 for further explanations. This sequence is related to A322269(6).
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.
The b(n) corresponding to each a(n) is A008366(n).
The first 32 terms are trivial: f=b, and then the product b*f naturally is a square modulo everything.
LINKS
EXAMPLE
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.
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(6) returns this sequence.
\\ Similarly, sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(5) returns A322274.
CROSSREFS
Sequence in context: A054484 A054796 A352379 * A008366 A126769 A092216
KEYWORD
nonn,fini,full
AUTHOR
Hans Ruegg, Dec 01 2018
STATUS
approved

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Last modified May 21 03:57 EDT 2024. Contains 372720 sequences. (Running on oeis4.)