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 A306508 Squarefree numbers that have fully composite idempotent factorizations. 3
 210, 462, 570, 1155, 1302, 1330, 1365, 1785, 2210, 2310, 2730, 3003, 3410, 3710, 3990, 4305, 4515, 4758, 4810, 5005, 5187, 5474, 5610, 5642, 6006, 6105, 6118, 6270, 6510, 6622, 6630, 7410, 7770, 8265, 8385, 8463, 8645, 9282, 9471, 9870, 10010, 10101, 10230, 10374, 10545, 10582 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Fully composite idempotent factorizations are bipartite factorizations n=p*q such that p and q are composite numbers with the property that for any b in Z_n, b^(k(p-1)(q-1)+1) is congruent to b mod n for any integer k >= 0. Idempotent factorizations have the property that p and q generate correctly functioning RSA keys, even if one or both are composite. 2730 has more than one fully composite idempotent factorization (10*273, 21*130). It is the smallest positive integer with that property. 7770 and 8463 are similar. LINKS Barry Fagin, Table of n, a(n) for n = 1..63737 B. Fagin, Idempotent Factorizations of Square-Free Integers, Information 2019, 10(7), 232. EXAMPLE 210=10*21, 462=22*21, 570=10*57, 1155=21*55, 1302=6*217, 1330=10*133, 1365=15*91 and 1785=21*85 are the fully composite idempotent factorizations for the first eight terms. PROG (Python) for n in range(2, max_n):     factor_list = numbthy.factor(n)     numFactors = len(factor_list)     if numFactors <= 3:         continue     if not bsflib.is_composite_and_square_free_with_list(n, factor_list):         continue     fciFactorizations = bsflib.fullyCompositeIdempotentFactorizations(n, factor_list)     numFCIFs = len(fciFactorizations)     if numFCIPs > 0:         fcIdempotents += 1     print(n) (PARI) isokc(p, q, n) = (p != 1) && !isprime(p) && !isprime(q) && (frac((p-1)*(q-1)/lcm(znstar(n)[2])) == 0); isok(n) = {if (issquarefree(n) && omega(n) >= 3, my(d = divisors(n)); for (k=1, #d\2, if (isokc(d[k], n/d[k], n), return (1); ); ); ); } \\ Michel Marcus, Feb 22 2019 CROSSREFS Cf. A115957, A138636, A002322. Subsequence of A120944 (composite squarefree numbers). Subsequence of A306330 (composite squarefree numbers with idempotent factorizations). Sequence in context: A325991 A264664 A147571 * A254466 A235304 A121479 Adjacent sequences:  A306505 A306506 A306507 * A306509 A306510 A306511 KEYWORD nonn AUTHOR Barry Fagin, Feb 20 2019 STATUS approved

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Last modified January 20 14:24 EST 2020. Contains 331094 sequences. (Running on oeis4.)