

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
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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[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(4) returns this sequence.


CROSSREFS



KEYWORD

nonn,fini,full


AUTHOR



STATUS

approved



