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.

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

Offset: 0

Author

Antti Karttunen, Jan 02 2014

Keywords

• nonn

Comments

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, ...)

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..511
• Index entries for sequences operating on (or containing) GF(2)[X]-polynomials

Crossrefs

Odd nonprimes in A235034. A235039 is a subsequence.

The composite terms in A045544 (A004729) all occur also here.

Cf. also A019434, A048720, A235045, A235050, A115857, A115872.