A115857 -id:A115857 - cp's OEIS frontend

A115858 Odd terms in A115857.

1, 7, 13, 11, 25, 31, 21, 19, 49, 23, 29, 47, 41, 31, 37, 35, 97, 39, 61, 63, 105, 59, 53, 55, 81, 55, 93, 59, 73, 63, 69, 67, 193, 71, 77, 75, 89, 91, 93, 91, 113, 87, 109, 99, 105, 107, 101, 167, 161, 103, 117, 107, 121, 155, 157, 115, 145, 119, 157, 143, 137, 139

Offset: 1

Antti Karttunen, Feb 07 2006

nonn

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Bisection of A115857.

A065621 Reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n.

Original entry on oeis.org

1, 2, 7, 4, 13, 14, 11, 8, 25, 26, 31, 28, 21, 22, 19, 16, 49, 50, 55, 52, 61, 62, 59, 56, 41, 42, 47, 44, 37, 38, 35, 32, 97, 98, 103, 100, 109, 110, 107, 104, 121, 122, 127, 124, 117, 118, 115, 112, 81, 82, 87, 84, 93, 94, 91, 88, 73, 74, 79, 76, 69, 70, 67, 64, 193

Offset: 1

Marc LeBrun, Nov 07 2001

a(0)=0. The alternation is applied only to the nonzero bits and does not depend on the exponent of two. All integers have a unique reversing binary representation (see cited exercise for proof). Complement of A048724.
A permutation of the "odious" numbers A000069.
Write n-1 and 2n-1 in binary and add them mod 2; example: n = 6, n-1 = 5 = 101 in binary, 2n-1 = 11 = 1011 in binary and their sum is 1110 = 14, so a(6) = 14. - Philippe Deléham, Apr 29 2005
As already pointed out, this is a permutation of the odious numbers A000069 and A010060(A000069(n)) = 1, so A010060(a(n)) = 1; and A010060(A048724(n)) = 0. - Philippe Deléham, Apr 29 2005. Also a(n) = A000069(A003188(n - 1)).

			a(5) = 13 = 8 + 4 + 1 -> 8 - 4 + 1 = 5.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 178, (exercise 4.1. Nr. 27)

Reinhard Zumkeller, Table of n, a(n) for n = 1..8192

Cf. A003188, A065620, A048724, A072219, A073122.
Differs from A115857 for the first time at n=19, where a(19)=55, while A115857(19)=23. Cf. A104895, A115872, A114389, A114390, A105081.
Cf. A245471.

import Data.Bits (xor, (.&.))
a065621 n = n `xor` 2 * (n - n .&. negate n) :: Integer
-- Reinhard Zumkeller, Mar 26 2014

f[n_] := BitXor[n, 2 n + 1]; Array[f, 60, 0] (* Robert G. Wilson v, Jun 09 2010 *)

a(n)=if(n<2,1,if(n%2==0,2*a(n/2),2*a((n+1)/2)-2*(-1)^((n-1)/2)+1))

def a(n): return n^(2*(n - (n & -n))) # Indranil Ghosh, Jun 04 2017

def A065621(n): return n^ (n&~-n)<<1 # Chai Wah Wu, Jun 29 2022

a(n) = if n=0 or n=1 then n else b+2*a(b+(1-2*b)*n)/2) where b is the least significant bit in n.
a(n) = n XOR 2 (n - (n AND -n)).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = 2*a(n+1) - 2(-1)^n + 1. - Ralf Stephan, Aug 20 2003
a(n) = A048724(n-1) - (-1)^n. - Ralf Stephan, Sep 10 2003
a(n) = Sum_{k=0..n} (1-(-1)^round(-n/2^k))/2*2^k. - Benoit Cloitre, Apr 27 2005
Closely related to Gray codes in another way: a(n) = 2 * A003188(n-1) + (n mod 2); a(n) = 4 * A003188((n-1) div 2) + (n mod 4). - Matt Erbst (matt(AT)erbst.org), Jul 18 2006 [corrected by Peter Munn, Jan 30 2021]
a(n) = n XOR 2(n AND NOT -n). - Chai Wah Wu, Jun 29 2022
a(n) = A003188(2n-1). - Friedjof Tellkamp, Jan 18 2024

More terms from Ralf Stephan, Sep 08 2003

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

Antti Karttunen, Jan 22 2014

"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.

			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.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

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).

A234742(n)=factorback(subst(lift(factor(Mod(1,2)*Pol(binary(n)))),x,2)) \\ M. F. Hasler, Feb 18 2014, corrected Andrew Howroyd, Aug 01 2018

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).

A115872 Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 4, 3, 6, 1, 5, 4, 7, 2, 7, 6, 5, 12, 3, 14, 3, 7, 6, 14, 4, 15, 6, 7, 8, 7, 15, 5, 28, 7, 14, 1, 9, 8, 24, 6, 30, 12, 15, 2, 15, 10, 9, 28, 7, 31, 14, 28, 3, 30, 7, 11, 10, 30, 8, 56, 15, 30, 4, 31, 14, 3, 12, 11, 31, 9, 60, 24, 31, 5, 60, 15, 6, 3, 13, 12, 48, 10, 62, 28, 56, 6, 62, 28, 12, 6, 5, 14, 13, 51, 11, 63, 30, 60, 7, 63, 30, 15, 7, 10, 7

Offset: 1

Antti Karttunen, Feb 07 2006

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
Square array is read by descending antidiagonals, as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Rows at positions 2^k are 1, 2, 3, ..., (A000027). Row 2n is equal to row n.
Numbers on each row give a subset of positions of zeros at the corresponding row of A284270. - Antti Karttunen, May 08 2019

			Fifteen initial terms of rows 1 - 19 are listed below:
   1:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   2:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   3:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
   4:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   5:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
   6:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
   7:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
   8:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
   9: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  10:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
  11:  3,  6, 12,  15,  24,  27,  30,  31,  48,  51,  54,  60,  62,  63,  96, ...
  12:  3,  6,  7,  12,  14,  15,  24,  28,  30,  31,  48,  51,  56,  60,  62, ...
  13:  5, 10, 15,  20,  21,  30,  31,  40,  42,  45,  47,  60,  61,  62,  63, ...
  14:  7, 14, 15,  28,  30,  31,  56,  60,  62,  63, 112, 120, 124, 126, 127, ...
  15: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  16:  1,  2,  3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,  15, ...
  17: 31, 62, 63, 124, 126, 127, 248, 252, 254, 255, 496, 504, 508, 510, 511, ...
  18: 15, 30, 31,  60,  62,  63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
  19:  7, 14, 28,  31,  56,  62,  63, 112, 119, 124, 126, 127, 224, 238, 248, ...

Transpose: A114388. First column: A115873.
Cf. A048720, A065621, A115857, A115871, A325565, A325566, A325567, A325568, A325570, A325571.
Cf. also arrays A277320, A277810, A277820, A284270.
A few odd-positioned rows: row 1: A000027, Row 3: A048717, Row 5: A115770 (? Checked for all values less than 2^20), Row 7: A115770, Row 9: A115801, Row 11: A115803, Row 13: A115772, Row 15: A115801 (? Checked for all values less than 2^20), Row 17: A115809, Row 19: A115874, Row 49: A114384, Row 57: A114386.

X[a_, b_] := Module[{A, B, C, x},
     A = Reverse@IntegerDigits[a, 2];
     B = Reverse@IntegerDigits[b, 2];
     C = Expand[
        Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*
        Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];
     PolynomialMod[C, 2] /. x -> 2];
T[n_, k_] := Module[{x = BitXor[n-1, 2n-1], k0 = k},
     For[i = 1, True, i++, If[n*i == X[x, i],
     If[k0 == 1, Return[i], k0--]]]];
Table[T[n-k+1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2022 *)

up_to = 120;
A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A115872sq(n, k) = { my(x = A065621(n)); for(i=1,oo,if((n*i)==A048720(x,i),if(1==k,return(i),k--))); };
A115872list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A115872sq(col,(a-(col-1))))); (v); };
v115872 = A115872list(up_to);
A115872(n) = v115872[n]; \\ (Slow) - Antti Karttunen, May 08 2019

Example section added and the data section extended up to n=105 by Antti Karttunen, May 08 2019

A235041 Factorization-preserving bijection from nonnegative integers to GF(2)[X]-polynomials, version which fixes the elements that are irreducible in both semirings.

Original entry on oeis.org

0, 1, 2, 3, 4, 25, 6, 7, 8, 5, 50, 11, 12, 13, 14, 43, 16, 55, 10, 19, 100, 9, 22, 87, 24, 321, 26, 15, 28, 91, 86, 31, 32, 29, 110, 79, 20, 37, 38, 23, 200, 41, 18, 115, 44, 125, 174, 47, 48, 21, 642, 89, 52, 117, 30, 227, 56, 53, 182, 59, 172, 61, 62, 27, 64

Offset: 0

Antti Karttunen, Jan 02 2014

nonn

Like A091202 this is a factorization-preserving isomorphism from integers to GF(2)[X]-polynomials. The latter are encoded in the binary representation of n like this: n=11, '1011' in binary, stands for polynomial x^3+x+1, n=25, '11001' in binary, stands for polynomial x^4+x^3+1. However, this version does not map the primes (A000040) straight to the irreducible GF(2)[X] polynomials (A014580), but instead fixes the intersection of those two sets (A091206), and maps the elements in their set-wise difference A000040 \ A014580 (= A091209) in numerical order to the set-wise difference A014580 \ A000040 (= A091214).
The composite values are defined by the multiplicativity. E.g., we have a(3n) = A048724(a(n)) and a(3^n) = A001317(n) for all n.
This map satisfies many of the same identities as A091202, e.g., we have A000005(n) = A091220(a(n)), A001221(n) = A091221(a(n)), A001222(n) = A091222(a(n)) and A008683(n) = A091219(a(n)) for all n >= 1.

			Here (t X u) = A048720(t,u):
a(2)=2, a(3)=3 and a(7)=7, as 2, 3 and 7 are all in A091206.
a(4) = a(2*2) = a(2) X a(2) = 2 X 2 = 4.
a(9) = a(3*3) = a(3) X a(3) = 3 X 3 = 5.
a(5) = 25, as 5 is the first term of A091209 and 25 is the first term of A091214.
a(10) = a(2*5) = a(2) X a(5) = 2 X 25 = 50.
Similarly, a(17) = 55, as 17 is the second term of A091209 and 55 is the second term of A091214.
a(21) = a(3*7) = a(3) X a(7) = 3 X 7 = 9.

Inverse: A235042. Fixed points: A235045.
Cf. A010051, A048720, A091225, A091214, A235043, A048724, A115857, A115872.
Similar cross-multiplicative permutations: A091202, A091204, A106442, A106444, A106446.

a(0)=0, a(1)=1, a(p) = p for those primes p whose binary representations encode also irreducible GF(2)[X]-polynomials (i.e., p is in A091206), and for the rest of the primes q (those whose binary representation encode composite GF(2)[X]-polynomials, i.e., q is in A091209), a(q) = A091214(A235043(q)), and for composite natural numbers, a(p * q * r * ...) = a(p) X a(q) X a(r) X ..., where p, q, r, ... are primes and X stands for the carryless multiplication (A048720) of GF(2)[X] polynomials encoded as explained in the Comments section.

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

Antti Karttunen, Jan 02 2014

nonn

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, ...)

			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.

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.

A115871 Table listing for each n (in descending order) all m's <= n, such that there exists nonzero solutions to a cross-domain congruence m*i = n X i.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 3, 8, 9, 10, 11, 7, 12, 13, 5, 14, 6, 15, 16, 17, 18, 19, 15, 20, 21, 13, 22, 14, 23, 19, 15, 24, 25, 9, 26, 10, 27, 15, 28, 12, 29, 21, 13, 30, 31, 27, 11, 32, 33, 34, 35, 31, 36, 37, 29, 38, 30, 39, 35, 40, 41, 25, 42, 26, 43, 27, 44, 28, 45, 46, 38, 30, 47, 27

Offset: 1

Antti Karttunen, Feb 07 2006

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

			Row n has A115861(n)+1 elements: 1; 2; 3; 4; 5; 6; 7,3; 8; 9; 10; 11,7; 12; 13,5; 14,6; 15; etc.

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

Cf. A115857, A115859, A115861, A115869, A115872.

A115859 Largest natural number m < n, such that there exists nonzero solutions to a cross-domain congruence m*i = n X i, zero if no such integer exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 7, 0, 5, 6, 0, 0, 0, 0, 15, 0, 13, 14, 19, 0, 9, 10, 15, 12, 21, 0, 27, 0, 0, 0, 31, 0, 29, 30, 35, 0, 25, 26, 27, 28, 0, 38, 27, 0, 17, 18, 0, 20, 45, 30, 51, 24, 25, 42, 55, 0, 45, 54, 59, 0, 0, 0, 63, 0, 61, 62, 67, 0, 57, 58, 71, 60, 69, 70, 63, 0

Offset: 1

Antti Karttunen, Feb 07 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

a(2n) = 2*a(n). Bisection A115860 gives the terms at odd positions. Differs from A115869 for the first time at n=23, where a(23)=19, while A115869(23)=15. Cf. A115857, A115861, A115871.

A115869 Smallest natural number m < n, such that there exists nonzero solutions to a cross-domain congruence m*i = n X i, zero if no such integer exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 7, 0, 5, 6, 0, 0, 0, 0, 15, 0, 13, 14, 15, 0, 9, 10, 15, 12, 13, 0, 11, 0, 0, 0, 31, 0, 29, 30, 35, 0, 25, 26, 27, 28, 0, 30, 23, 0, 17, 18, 0, 20, 45, 30, 19, 24, 25, 26, 23, 0, 21, 22, 27, 0, 0, 0, 63, 0, 61, 62, 63, 0, 57, 58, 59, 60, 61, 70, 55, 0

Offset: 1

Antti Karttunen, Feb 07 2006

nonn

Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

Index entries for sequences defined by congruent products between domains N and GF(2)[X]

a(2n) = 2*a(n). Bisection A115870 gives the terms at odd positions. Differs from A115859 for the first time at n=23, where a(23)=15, while A115859(23)=19. Cf. A115857, A115861, A115871.

cp's OEIS Frontend

A115858 Odd terms in A115857.

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A065621 Reversing binary representation of n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives n.

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

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A115872 Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.

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A235041 Factorization-preserving bijection from nonnegative integers to GF(2)[X]-polynomials, version which fixes the elements that are irreducible in both semirings.

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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.

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A115871 Table listing for each n (in descending order) all m's <= n, such that there exists nonzero solutions to a cross-domain congruence m*i = n X i.

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A115859 Largest natural number m < n, such that there exists nonzero solutions to a cross-domain congruence m*i = n X i, zero if no such integer exists.

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A115869 Smallest natural number m < n, such that there exists nonzero solutions to a cross-domain congruence m*i = n X i, zero if no such integer exists.