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 1-6 of 6 results.

A234742 Product of the binary encodings of the irreducible factors (with multiplicity) of the polynomial over GF(2) whose encoding is n.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 7, 8, 21, 18, 11, 12, 13, 14, 27, 16, 81, 42, 19, 36, 49, 22, 39, 24, 25, 26, 63, 28, 33, 54, 31, 32, 93, 162, 91, 84, 37, 38, 99, 72, 41, 98, 75, 44, 189, 78, 47, 48, 77, 50, 243, 52, 57, 126, 55, 56, 117, 66, 59, 108, 61, 62, 147, 64, 441
Offset: 0

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Author

Antti Karttunen, Jan 22 2014

Keywords

Comments

"Product" refers to the ordinary multiplication of integers.
Differs from A235042 and A236837 for the first time at n=25, where a(n)=25, while A235042(25)=5 and A236837(25)=0. Thus A234741(A234742(n)) = n up to n=24.
a(n) >= n. [All terms of the table A061858 are nonnegative as the product of multiplying two numbers with carries is never less than when multiplying them without carries.]
Specifically, for all n, a(A091209(n)) > A091209(n).
a(A091209(n)) is always composite and, by the above inequality, larger than A091209(n), which implies that none of the terms of A091209 occur in this sequence. Cf. also A236844.
Starting with various terms (primes) in A235033 and iterating the map A234742, we get 5 -> 9 -> 21 -> 49 -> 77 -> 177 -> 333 = a(333).
Another example: 17 -> 81 -> 169 -> 309 -> 721 = a(721).
Does every chain of such iterations eventually reach a fixed point? (One of the terms of A235035.) Or do some of them manage to avoid such "traps" indefinitely? (Note how the terms of A235035 seem to get rarer, but only rather slowly.)
Starting from 23, we get the sequence: 23, 39, 99, 279, 775, 1271, 3003, 26411, 45059, ... which reaches its fixed point, 3643749709604450870616156947649219, after 55 iterations. - M. F. Hasler, Feb 18 2014. [This is now sequence A244323. See also A260729, A260735 and A260441.] - Antti Karttunen, Aug 05 2015
Note also that when coming backwards from some term of such a chain by iterating A234741, we may not necessarily end at the same term we started from.

Examples

			3 has binary representation '11', which encodes the polynomial X + 1, which is irreducible in GF(2)[X], so the result is just a(3)=3.
5 has binary representation '101' which encodes the polynomial X^2 + 1, which is reducible in the polynomial ring GF(2)[X], factoring as (X+1)(X+1), i.e., 5 = A048720(3,3), as 3 ('11' in binary) encodes the polynomial (X+1), irreducible in GF(2)[X]. 3*3 = 9, thus a(5)=9.
9 has binary representation '1001', which encodes the polynomial X^3 + 1, which factors (in GF(2)[X]!) as (X+1)(X^2+X+1), i.e., 9 = A048720(3,7) (7, '111' in binary, encodes the other factor polynomial X^2+X+1). 3*7 = 21, thus a(9)=21.
25 has binary representation '11001', which encodes the polynomial X^4 + X^3 + 1, which is irreducible in GF(2)[X], so the result is just a(25)=25.

Links

Crossrefs

A235035 gives the k for which a(k)=k.
A236853(n) gives the number of times n occurs in this sequence.
A236842 gives the same sequence sorted and with duplicates removed, A236844 gives the numbers that do not occur here, A236845 gives numbers that occur more than once, A236846 the least inverse and A236847 the greatest inverse. A236850 gives such k that a(k) = A236837(k).
Cf. A014580, A048720, A234741, A236379, A236837, A236851 & A236852, A115857, A115872.
Cf. also A260712, A260713, A260716 and A244323, A260729, A260735, A260441 (iterations starting from various terms of A236844).

Programs

Formula

To compute a(n): factor the polynomial over GF(2) encoded by n, into its irreducible factors; in other words, find a unique multiset of terms i, j, ..., k (not necessarily distinct) from A014580 for which i x j x ... x k = n, where x stands for the carryless multiplication A048720. Then a(n) = i*j*...*k is the product of those terms with ordinary multiplication. Because of the effect of the carry-bits in the latter, the result is always greater than or equal to n, so we have a(n) >= n for all n.
a(2n) = 2*a(n).
a(A235035(n)) = A235035(n).
A236379(n) = a(n) - n.
For all n, a(n) >= A236837(n).

A235035 Numbers n for which A234742(n) = n: numbers n whose binary representation encodes a GF(2)[X]-polynomial such that when its irreducible factors are multiplied together as ordinary integers (with carry-bits), the result is n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 19, 22, 24, 25, 26, 28, 31, 32, 37, 38, 41, 44, 47, 48, 50, 52, 55, 56, 59, 61, 62, 64, 67, 73, 74, 76, 82, 87, 88, 91, 94, 96, 97, 100, 103, 104, 109, 110, 111, 112, 115, 117, 118, 122, 123, 124, 128, 131, 134, 137
Offset: 1

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Author

Antti Karttunen, Jan 02 2014

Keywords

Links

Crossrefs

Gives the positions of zeros in A236379, i.e., n such that A234742(n) = n.
An intersection with A235034 gives A235032. Contains A014580 as a subsequence.
Cf. A235036, A235032-A235033, A234742.

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.

A236835 Numbers that occur in more than one way as results of "downward" remultiplication (N -> GF(2)[X]) of some number.

Original entry on oeis.org

5, 10, 15, 17, 20, 23, 27, 29, 30, 34, 39, 40, 45, 46, 51, 53, 54, 57, 58, 60, 65, 68, 71, 75, 78, 80, 83, 85, 90, 92, 95, 99, 101, 102, 105, 106, 107, 108, 113, 114, 116, 119, 120, 127, 129, 130, 135, 136, 139, 141, 142, 147, 150, 151, 153, 156, 160, 163, 165
Offset: 1

Views

Author

Antti Karttunen, Jan 31 2014

Keywords

Comments

Numbers that occur more than once in A234741.

Examples

			5 occurs here, because it occurs in A234741 both as A234741(5)=5 and A234741(9)=5, as A048720(3,3)=5.
43 do not occur here, as although it is a term of A091209, it only occurs at A234741(43) as it cannot be obtained by other means as a carryless product than as 43 = A048720(3,25).

Links

Crossrefs

Positions of terms larger than one in A236833.
Cf. A236833, A236834. A091209 is NOT a subsequence.

Formula

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

A236380 Difference between value of n, when remultiplied "upward", from GF(2)[X] to N, and when remultiplied "downward", from N to GF(2)[X]: a(n) = A234742(n) - A234741(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 0, 0, 16, 8, 0, 0, 0, 0, 12, 0, 64, 32, 0, 16, 40, 0, 16, 0, 8, 0, 48, 0, 4, 24, 0, 0, 64, 128, 64, 64, 0, 0, 76, 32, 0, 80, 32, 0, 172, 32, 0, 0, 56, 16, 192, 0, 4, 96, 16, 0, 64, 8, 0, 48, 0, 0, 120, 0, 384, 128, 0, 256, 64, 128, 112, 128, 0, 0, 300, 0, 128, 152, 96, 64, 152, 0, 148, 160, 644, 64
Offset: 0

Views

Author

Antti Karttunen, Jan 24 2014

Keywords

Comments

All terms are divisible by 4.
a(n) = 0 iff both A236378(n) and A236379(n) are zero, or in other words, iff A234741(n)=n and A234742(n)=n, which means that A235032 gives all such n, that a(n) = 0.

Links

Crossrefs

A235032 gives the positions of zeros, A235033 the positions of nonzeros.
Cf. also A234741, A234742, A236378, A236379, A236851, A061858.

Formula

a(n) = A234742(n) - A234741(n).
a(n) = A236378(n) + A236379(n).

A236378 Difference between n and the result obtained when n is remultiplied from N to GF(2)[X]: a(n) = n - A234741(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 12, 0, 0, 0, 8, 0, 12, 0, 0, 0, 0, 0, 4, 0, 8, 16, 0, 0, 16, 0, 0, 24, 0, 0, 28, 0, 0, 0, 28, 16, 0, 0, 0, 24, 16, 0, 4, 0, 0, 0, 0, 0, 36, 0, 8, 8, 0, 0, 12, 16, 0, 32, 0, 0, 24, 0, 28, 32, 0, 0, 64, 0, 0, 48, 0, 0, 48, 0, 0, 56, 56, 0, 60
Offset: 0

Views

Author

Antti Karttunen, Jan 24 2014

Keywords

Comments

a(n) is the difference of n and the number obtained when the prime divisors of n are multiplied together in such a way that the carry-bits from intermediate products are discarded, as in A048720.
All terms are divisible by 4.

Links

Crossrefs

Cf. A235034 (gives the positions of zeros).
Cf. A234741, A236379, A236380, A061858.
Showing 1-6 of 6 results.

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