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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

Barry Fagin, All n < 2^27 and their fully composite idempotent factorizations

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.)