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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

%I

%S 1,11,13,17,19,23,29,31,37,41,43,47,53,59,61,43,71,73,79,83,41,73,101,

%T 103,107,109,113,1,127,59,113,19,47,29,79,13,43,47,1,173,11,61,283,71,

%U 193,53,31,41,211,29,103,83,61,113,71,241,127,59,37,17,23

%N 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.

%C See sequence A322269 for further explanations. This sequence is related to A322269(4).

%C 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.

%C The b(n) corresponding to each a(n) is A008364(n).

%C The first 15 terms are trivial: f=b, and then the product b*f naturally is a square modulo everything.

%H Hans Ruegg, <a href="/A322273/b322273.txt">Table of n, a(n) for n = 1..192</a>

%e The 16th number coprime to 840 is 67. a(16) is 43, because 43 is the smallest prime with 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.

%o (PARI)

%o QresCode(n, nPrimes) = {

%o code = bitand(n,7)>>1;

%o for (j=2, nPrimes,

%o x = Mod(n,prime(j));

%o if (issquare(x), code += (1<<j));

%o );

%o return (code);

%o }

%o QCodeArray(n) = {

%o totalEntries = 1<<(n+1);

%o f = vector(totalEntries);

%o f[totalEntries-3] = 1; \\ 1 has always the same code: ...111100

%o counter = 1;

%o forprime(p=prime(n+1), +oo,

%o code = QresCode(p, n);

%o if (f[code+1]==0,

%o f[code+1]=p;

%o counter += 1;

%o if (counter==totalEntries, return(f));

%o )

%o )

%o }

%o sequence(n) = {

%o f = QCodeArray(n);

%o primorial = prod(i=1, n, prime(i));

%o entries = eulerphi(4*primorial);

%o a = vector(entries);

%o i = 1;

%o forstep (x=1, 4*primorial-1, 2,

%o if (gcd(x,primorial)==1,

%o a[i] = f[QresCode(x, n)+1];

%o i += 1;

%o );

%o );

%o return(a);

%o }

%o \\ sequence(4) returns this sequence.

%o \\ sequence(2) returns A322271, sequence(3) returns A322272, ... sequence(6) returns A322275.

%Y Cf. A322269, A322271, A322272, A322274, A322275, A008364.

%K nonn,fini,full

%O 1,2

%A _Hans Ruegg_, Dec 01 2018

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)