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A322273 Smallest multiplication factors f, prime or 1, for all b (mod 840), coprime to 840 (= 4*7#), so that b*f is a nonzero square mod 8, mod 3, mod 5, and mod 7. 7
1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 43, 71, 73, 79, 83, 41, 73, 101, 103, 107, 109, 113, 1, 127, 59, 113, 19, 47, 29, 79, 13, 43, 47, 1, 173, 11, 61, 283, 71, 193, 53, 31, 41, 211, 29, 103, 83, 61, 113, 71, 241, 127, 59, 37, 17, 23 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
See sequence A322269 for further explanations. This sequence is related to A322269(4).
The sequence is periodic, repeating itself after phi(840) = 192 terms. Its largest term is 311, which is A322269(4). In order to satisfy the conditions, both f and b must be coprime to 840. Otherwise, the product would be zero mod a prime <= 7.
The b(n) corresponding to each a(n) is A008364(n).
The first 15 terms are trivial: f=b, and then the product b*f naturally is a square modulo everything.
LINKS
EXAMPLE
The 16th number coprime to 840 is 67. a(16) is 43, because 43 is the smallest prime by which we can multiply 67, so that the product (67*43 = 2881) is a square mod 8, mod 2, mod 3, mod 5, and mod 7.
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(4) returns this sequence.
\\ sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.
CROSSREFS
Sequence in context: A084374 A063193 A056758 * A096489 A008364 A140461
KEYWORD
nonn,fini,full
AUTHOR
Hans Ruegg, Dec 01 2018
STATUS
approved

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Last modified May 29 05:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)