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A235040 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.

%I A235040 #18 Feb 05 2014 10:54:20
%S A235040 1,15,51,85,95,111,119,123,187,219,221,255,335,365,411,447,485,511,
%T A235040 629,655,685,697,771,831,879,959,965,1011,1139,1241,1285,1405,1535,
%U A235040 1563,1649,1731,1779,1799,1923,1983,2005,2019,2031,2045,2227,2605,2735,2815,2827
%N A235040 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.
%C A235040 Note: Start indexing from n=1 if you want just composite numbers. a(0)=1 is the only nonprime, noncomposite in this list.
%C A235040 The first term with three prime divisors is a(11) = 255 = 3*5*17.
%C A235040 The next terms with three prime divisors are
%C A235040   255, 3855, 13107, 21845, 24415, 28527, 30583, 31215, 31611, 31695, 32691, 48059, 56283, 56797, 61935, 65365, 87805, 98005, ...
%C A235040 Of these 24415 (= 5*19*257) is the first one with at least one prime factor that is not a Fermat prime (A019434).
%C A235040 The first term with four prime divisors is a(427) = 65535 = 3*5*17*257.
%C A235040 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, ...)
%H A235040 Antti Karttunen, <a href="/A235040/b235040.txt">Table of n, a(n) for n = 0..511</a>
%H A235040 <a href="/index/Ge#GF2X">Index entries for sequences operating on (or containing) GF(2)[X]-polynomials</a>
%e A235040 15 = 3*5. When these factors (with binary representations '11' and '101') are multiplied as:
%e A235040    101
%e A235040   1010
%e A235040   ----
%e A235040   1111 = 15
%e A235040 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.
%o A235040 (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o A235040 (define A235040 (MATCHING-POS 0 1 (lambda (n) (and (odd? n) (not (prime? n)) (= n (reduce A048720bi 1 (ifactor n)))))))
%Y A235040 Odd nonprimes in A235034. A235039 is a subsequence.
%Y A235040 The composite terms in A045544 (A004729) all occur also here.
%Y A235040 Cf. also A019434, A048720, A235045, A235050, A115857, A115872.
%K A235040 nonn
%O A235040 0,2
%A A235040 _Antti Karttunen_, Jan 02 2014