The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.


(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A322269 a(n) is the largest minimal prime P such that, for any odd number b, the product P*b is a nonzero square modulo 8 and modulo each of the first n primes. 7
7, 23, 83, 311, 1873, 3583, 12289, 33049, 67369, 174241, 552841, 1010881, 3267289, 7921489, 12537719, 30706079, 56988649, 108345169, 328583161, 880051561, 1644946249 (list; graph; refs; listen; history; text; internal format)
When factoring a number b using the quadratic sieve, it can be practical to multiply b by a certain factor f so that the product f*b is a square modulo several small primes. It is desirable that f be prime, because the prime factors of f cannot be used in the factor base of the quadratic sieve.
To find such an f for a given b and the first n primes, it must be checked whether b is a square or not, modulo each of these primes. Then f is the smallest prime (or 1) which satisfies the same conditions, modulo each of these primes.
Letting p=prime(n), an f can be found for each of the possible values of b (mod p#, the primorial of p), coprime to p#. (Actually we are using a period of 4*(p#), because instead of mod 2 we check for mod 8.) a(n) is the largest of all these values of f.
8 was chosen instead of 2, because there is a unique quadratic residue (mod 8), i.e., 1, for all odd numbers.
Sequences A322271 to A322275 are separate listings for the sequences of all f, corresponding to n=2 to 6, which illustrate the idea further.
For finding the full sequences of all f, instead of checking all b mod 4*(p#), it is more practical to check all prime numbers (and also 1) in order, whether they are suitable as an f or not. Each prime receives a "code" of Boolean flags which indicate whether it is a square or not, modulo each of the first n primes. If it is the first prime with this specific "code", then every value of b mod 4*(p#) which has the same "code" is assigned this prime as its f. This process is repeated until all possible "codes" have an f assigned. (The flag for mod 8, instead of only signaling "is (not) a square", has four different values: 1, 3, 5, and 7.)
A322270(n) is the code corresponding to a(n).
In order to satisfy the conditions, both f and b must be coprime to p#, i.e., f must either be 1 or greater than prime(n).
For n=3, we want the product to be a square mod 8, mod 2, mod 3 and mod 5. The corresponding products b*f are, for all b < 120 and coprime to 120:
1*1, 7*7, 11*11, 13*13, 17*17, 19*19, 23*23, 29*29, 31*31, 37*13, 41*41, 43*43, 47*23, 49*1, 53*53, 59*11, 61*61, 67*43, 71*71, 73*73, 77*53, 79*31, 83*83, 89*41, 91*19, 97*73, 101*29, 103*7, 107*83, 109*61, 113*17, 119*71. (See A322272.)
The largest f in this set is 83 (associated with b=83 and b=107). Therefore a(3) = 83.
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);
a322269(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,
counter += 1;
if (counter==totalEntries, return(p));
Binary codes as described above are given in A322270.
Sequences for all f associated with a specific n are given in A322271 (n=2), A322272 (n=3), A322273 (n=4), A322274 (n=5), and A322275 (n=6).
Sequence in context: A086908 A093069 A341665 * A303890 A003540 A063793
Hans Ruegg, Dec 01 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 28 12:54 EDT 2024. Contains 372913 sequences. (Running on oeis4.)