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.

Showing 1-4 of 4 results.

A236844 Numbers that do not occur as results of "upward" remultiplication (GF(2)[X] -> N) of any number; numbers not present in A234742.

Original entry on oeis.org

5, 10, 15, 17, 20, 23, 29, 30, 34, 35, 40, 43, 45, 46, 51, 53, 58, 60, 65, 68, 69, 70, 71, 79, 80, 83, 85, 86, 89, 90, 92, 95, 101, 102, 105, 106, 107, 113, 116, 119, 120, 125, 127, 129, 130, 135, 136, 138, 139, 140, 142, 149, 151, 153, 155, 158, 159, 160, 161
Offset: 1

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Author

Antti Karttunen, Jan 31 2014

Keywords

Comments

Numbers that do not occur in A234742 (A236842).
This is a subsequence of A236848, thus all terms are divisible by at least one such prime which is reducible as polynomial over GF(2) (i.e. one of the primes in A091209).
A236835(7)=27 is the first member of A236835 which does not occur here. a(12)=43 is the first term here which does not occur in A236835.

Links

Crossrefs

Complement: A236842.
A setwise difference of A236848 and A236849.
A091209 is a subsequence.
Positions of zeros in A236853, A236846, A236847 and A236862.
Cf. A236845.
Cf. also A236834.

Formula

For all n, A236379(a(n)) > 0.

A236860 After 0 and 1, numbers n all of whose prime divisors encode an irreducible polynomial over GF(2) (are terms of A091206).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 21, 22, 24, 26, 27, 28, 31, 32, 33, 36, 37, 38, 39, 41, 42, 44, 47, 48, 49, 52, 54, 56, 57, 59, 61, 62, 63, 64, 66, 67, 72, 73, 74, 76, 77, 78, 81, 82, 84, 88, 91, 93, 94, 96, 97, 98, 99, 103, 104, 108, 109
Offset: 1

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Author

Antti Karttunen, Mar 08 2014

Keywords

Comments

After 0 and 1, positive integers which are products of p * q * ... * r, where p, q, ..., r are terms of A091206.
Also fixed points of A236852(n). Proof: if k is a term of this sequence, the operation described in A236852 reduces to an identity operation. On the other hand, if k is not a term of this sequence, then it has at least one prime divisor which is reducible in polynomial ring GF(2)[X], which is thus "broken" by A236852 (A234742) to two or more separate factors (either prime or not), and because the original factor was prime, and N is a unique factorization domain, the new product computed over the new set of factors (with one or more "broken" pieces) cannot be equal to the original k. (Compare this to how primes are "broken" in a similar way in A235027, also A235145.)
Note: This sequence is not equal to all n for which A234741(n) = A236846(n). The first counterexample occurs at a(325) = 741 (= 3*13*19) for which we have: A236846(741) = 281 (= 3 x 247 = 3 x (13*19)) while A234741(741) = 329 (= 3 x 13 x 19). Contrast this with the behavior of the "dual sequence" A236850, where the corresponding property holds.

Links

Crossrefs

Complement: A236848.
Subsequence of A236842.
Fixed points of A236852.
Cf. A091206, A236850.

Programs

  • PARI
    isp(p) = polisirreducible( Mod(1, 2) * Pol(binary(p))); \\ A091206
isok(n) = if ((n==0), 1 , my(f=factor(n)); prod(k=1, #f~, isp(f[k,1])) != 0); \\ Michel Marcus, Dec 22 2018

A236838 Numbers whose binary representation encodes a polynomial over GF(2) with the property that at least one of its irreducible factors is encoded by a composite number.

Original entry on oeis.org

25, 43, 50, 55, 79, 86, 87, 89, 91, 100, 110, 115, 117, 125, 133, 135, 143, 145, 149, 158, 159, 171, 172, 174, 178, 181, 182, 185, 200, 203, 209, 213, 220, 227, 230, 234, 235, 237, 247, 249, 250, 253, 263, 266, 267, 270, 279, 281, 285, 286, 290, 293, 298, 299
Offset: 1

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Author

Antti Karttunen, Feb 02 2014

Keywords

Comments

Numbers which are of the form A048720(a,A091214(b)) for some a, b.
In the range 1..10000 about half of the natural numbers seem to be in this set, and the terms are getting more frequent, although rather slowly. (Please see the graph.)

Examples

			25, in binary '11001', encodes polynomial x^4 + x^3 + 1, which is irreducible in polynomial ring GF(2)[X], but is composite in N, thus it is a term of this sequence.
43, in binary '101011', encodes polynomial x^5 + x^3 + x + 1, which factors as (x + 1)(x^4 + x^3 + 1), i.e., 43 = A048720(3,25), and the latter factor of these, encoded by 25, is a composite in N, thus 43 is a term of this sequence.

Links

Crossrefs

Disjoint union of A236834 and A236839.
Complement: A236850.
Cf. A236848, A048720, A091214.

Programs

  • Scheme
    (define A236838 (MATCHING-POS 1 1 (lambda (n) (any (lambda (p) (= 1 (A066247 p))) (GF2Xfactor n)))))

A236849 Numbers that occur as results of remultiplication (GF(2)[X] -> N) of some number and have at least one prime divisor encoding a reducible polynomial in ring GF(2)[X].

Original entry on oeis.org

25, 50, 55, 75, 87, 100, 110, 115, 145, 150, 165, 174, 175, 185, 200, 203, 213, 220, 225, 230, 253, 261, 275, 285, 290, 299, 300, 301, 319, 325, 330, 345, 348, 350, 355, 357, 370, 375, 385, 391, 395, 400, 406, 415, 425, 426, 435, 440, 445, 450, 460, 475, 477, 495, 505, 506, 515, 522, 525, 529, 535, 545, 550, 555
Offset: 1

Views

Author

Antti Karttunen, Jan 31 2014

Keywords

Comments

Terms of A236842 (A234742) that are divisible by at least one of the primes in A091209.
a(4)=75, is the first term here which does not occur in A236834. On the other hand, A236834(5)=91 is the first of its terms that does not occur here.

Links

Crossrefs

Setwise difference A236848 \ A236844, and also A236842 \ A236860.
Cf. A091209, A234742, A236842, A236848, A236834.
Showing 1-4 of 4 results.

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