cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A235034 Numbers whose prime divisors, when multiplied together without carry-bits (as encodings of GF(2)[X]-polynomials, with A048720), produce the original number; numbers for which A234741(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 51, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 96, 97, 101
Offset: 1

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Author

Antti Karttunen, Jan 02 2014

Keywords

Comments

If n is present, then 2n is present also, as shifting binary representation left never produces any carries.

Examples

			All primes occur in this sequence as no multiplication -> no need to add any intermediate products -> no carry bits produced.
Composite numbers like 15 are also present, as 15 = 3*5, and 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

Crossrefs

Gives the positions of zeros in A236378, i.e., n such that A234741(n) = n.
Intersection with A235035 gives A235032.
Other subsequences: A000040 (A091206 and also A091209), A045544 (A004729), A093641, A235040 (gives odd composites in this sequence), A235050, A235490.
Cf. A048720, A061858, A234741.

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.

Original entry on oeis.org

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

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Author

Antti Karttunen, Jan 02 2014

Keywords

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

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.

A235490 Numbers such that none of their prime factors share common 1-bits in the same bit-position and when added (or "ored" or "xored") together, yield a term of A000225 (a binary "repunit").

Original entry on oeis.org

1, 3, 7, 10, 26, 31, 58, 122, 127, 1018, 2042, 8186, 8191, 32762, 131071, 524287, 2097146, 8388602, 33554426, 1073741818, 2147483647, 2305843009213693951, 618970019642690137449562111, 39614081257132168796771975162, 162259276829213363391578010288127, 166153499473114484112975882535043066
Offset: 1

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Author

Antti Karttunen, Jan 22 2014

Keywords

Comments

a(1) = 1 is included on the grounds that it has no prime factors, thus A001414(1)=0, and 0 is one of the terms of A000225, marking the "repunit of length zero".
After 1, the sequence is a union of A000668 (Mersenne primes) and semiprimes of the form 2*A050415. The terms were constructed from the data given in those two entries.

Examples

			7 is included, because it is a prime, and repunit in base-2: '111'.
10 is included, as 10=2*5, and when we add 2 ('10' in binary) and 5 ('101' in binary), we also get 7 ('111' in binary), without producing any carries.

Crossrefs

Subsequence of A235050.
Cf. A000225, A000668, A050415, A001414, A072593, A178910, A227843.
Cf. also A235488, A230492, A235032, A050376.
Showing 1-3 of 3 results.

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