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 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)
 OFFSET 1,1 COMMENTS 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). LINKS Table of n, a(n) for n=1..21. EXAMPLE 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. PROG (PARI) QresCode(n, nPrimes) = { code = bitand(n, 7)>>1; for (j=2, nPrimes, x = Mod(n, prime(j)); if (issquare(x), code += (1<

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