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A306508 Squarefree numbers that have fully composite idempotent factorizations.

%I A306508 #46 Dec 09 2021 01:00:23
%S A306508 210,462,570,1155,1302,1330,1365,1785,2210,2310,2730,3003,3410,3710,
%T A306508 3990,4305,4515,4758,4810,5005,5187,5474,5610,5642,6006,6105,6118,
%U A306508 6270,6510,6622,6630,7410,7770,8265,8385,8463,8645,9282,9471,9870,10010,10101,10230,10374,10545,10582
%N A306508 Squarefree numbers that have fully composite idempotent factorizations.
%C A306508 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.
%C A306508 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.
%H A306508 Barry Fagin, <a href="/A306508/b306508.txt">Table of n, a(n) for n = 1..63737</a>
%H A306508 Barry Fagin, <a href="/A306508/a306508.txt"> All n < 2^27 and their fully composite idempotent factorizations</a>
%H A306508 Barry Fagin, <a href="https://doi.org/10.3390/info10070232">Idempotent Factorizations of Square-Free Integers</a>, Information 2019, 10(7), 232.
%H A306508 Barry Fagin, <a href="https://doi.org/10.3390/info12080305">Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers</a>, Information (2021) Vol. 12, No. 8, 305.
%e A306508 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.
%o A306508 (Python)
%o A306508 for n in range(2,max_n):
%o A306508     factor_list = numbthy.factor(n)
%o A306508     numFactors = len(factor_list)
%o A306508     if numFactors <= 3:
%o A306508         continue
%o A306508     if not bsflib.is_composite_and_square_free_with_list(n,factor_list):
%o A306508         continue
%o A306508     fciFactorizations = bsflib.fullyCompositeIdempotentFactorizations(n,factor_list)
%o A306508     numFCIFs = len(fciFactorizations)
%o A306508     if numFCIPs > 0:
%o A306508         fcIdempotents += 1
%o A306508     print(n)
%o A306508 (PARI) isokc(p, q, n) = (p != 1) && !isprime(p) && !isprime(q) && (frac((p-1)*(q-1)/lcm(znstar(n)[2])) == 0);
%o A306508 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
%Y A306508 Cf. A115957, A138636, A002322.
%Y A306508 Subsequence of A120944 (composite squarefree numbers).
%Y A306508 Subsequence of A306330 (composite squarefree numbers with idempotent factorizations).
%K A306508 nonn
%O A306508 1,1
%A A306508 _Barry Fagin_, Feb 20 2019