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A306812
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Maximally idempotent integers with three or more factors.
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2
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273, 455, 1729, 2109, 2255, 2387, 3367, 3515, 4433, 4697, 4921, 5673, 6643, 6935, 7667, 8103, 8723, 8729, 9139, 9455, 10235, 10787, 11543, 13237, 13505, 14497, 16211, 16385, 16523, 17507, 18031, 18907, 20033, 20801, 21437, 22649, 23579, 24583
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OFFSET
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1,1
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COMMENTS
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An integer n has an idempotent factorization n=pq if b^(k(p-1)(q-1)+1) is congruent to b mod n for any integer k >= 0 and any b in Z_n (see A306330). An integer is maximally idempotent if all its bipartite factorizations n=pq are idempotent.
There are 15506 maximally idempotent integers less than 2^30. 15189 have three factors, 315 have four, two have five. The smallest maximally idempotent integer with four factors is 63973=7*13*19*37, a Carmichael number. The two with five factors are 13*19*37*73*109 and 11*31*41*101*151. The smallest maximally idempotent integer with six factors is 11*31*41*61*101*151.
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LINKS
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EXAMPLE
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273 is the smallest maximally idempotent integer. Factorization is (3,7,13). Bipartite factorizations are (3,91), (7,39), (13,21). Lambda(273) = 12, (2*90),(6*38) and (12*20) are all divisible by 12, thus 273 is maximally idempotent. The same is true for 455 = 5*7*13. The next entry in the sequence, 1729=7*13*19, is a Carmichael number, but most Carmichael numbers are not maximally idempotent.
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PROG
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(Python)
## This uses a custom library of number theory functions and the numbthy library.
## Hopefully the names of the functions make the process clear.
for n in range(2, max_n):
factor_list = numbthy.factor(n)
numFactors = len(factor_list)
if numFactors <= 2: # skip primes and semiprimes
continue
if not bsflib.is_composite_and_square_free_with_list(n, factor_list):
continue
ipList = bsflib.idempotentPartitions(n, factor_list)
if len(ipList) == 2**(numFactors-1)-1:
print(n)
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CROSSREFS
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Subsequence of A120944 (composite squarefree numbers). Subsequence of A306330 (squarefree numbers that admit idempotent factorizations). Members of the sequence with >= 4 factors for a subsequence of A306508 (squarefree integers with fully composite idempotent factorizations).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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