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A235040
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After 1, composite odd numbers, whose prime divisors, when multiplied together without carry-bits (as codes for GF(2)[X]-polynomials, with A048720), yield the same number back.
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8
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1, 15, 51, 85, 95, 111, 119, 123, 187, 219, 221, 255, 335, 365, 411, 447, 485, 511, 629, 655, 685, 697, 771, 831, 879, 959, 965, 1011, 1139, 1241, 1285, 1405, 1535, 1563, 1649, 1731, 1779, 1799, 1923, 1983, 2005, 2019, 2031, 2045, 2227, 2605, 2735, 2815, 2827
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OFFSET
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0,2
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COMMENTS
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Note: Start indexing from n=1 if you want just composite numbers. a(0)=1 is the only nonprime, noncomposite in this list.
The first term with three prime divisors is a(11) = 255 = 3*5*17.
The next terms with three prime divisors are
255, 3855, 13107, 21845, 24415, 28527, 30583, 31215, 31611, 31695, 32691, 48059, 56283, 56797, 61935, 65365, 87805, 98005, ...
Of these 24415 (= 5*19*257) is the first one with at least one prime factor that is not a Fermat prime (A019434).
The first term with four prime divisors is a(427) = 65535 = 3*5*17*257.
The first terms which are not multiples of any Fermat prime are: 511, 959, 3647, 4039, 4847, 5371, 7141, 7231, 7679, 7913, 8071, 9179, 12179, ... (511 = 7*73, 959 = 7*137, ...)
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LINKS
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EXAMPLE
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15 = 3*5. When these factors (with binary representations '11' and '101') are multiplied as:
101
1010
----
1111 = 15
we see that the intermediate products 1*5 and 2*5 can be added together without producing any carry-bits (as they have no 1-bits in the same columns/bit-positions), so A048720(3,5) = 3*5 and thus 15 is included in this sequence.
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PROG
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(define A235040 (MATCHING-POS 0 1 (lambda (n) (and (odd? n) (not (prime? n)) (= n (reduce A048720bi 1 (ifactor n)))))))
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CROSSREFS
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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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