A266195 -id:A266195 - cp's OEIS frontend

A266186 a(n) = A266196(A000079(n)); indices of powers of 2 in A266195.

1, 2, 4, 7, 12, 16, 25, 42, 50, 82, 104, 116, 201, 227, 243, 455, 477, 517, 1035, 1093, 1155, 1217, 2599, 2695, 4377, 4491, 4773, 4947, 5137, 13409, 13537, 14125, 14299, 14631, 15123, 34005, 34447, 34781, 36017, 36867, 37289, 37913, 155106, 155700, 157254

Offset: 0

Antti Karttunen, Dec 26 2015

Very likely also the positions of records in A264982.

Rémy Sigrist, C++ program

Cf. A000079, A264982, A266195, A266196, A266187.

a(n) = A266196(A000079(n)).

More terms from Rémy Sigrist, Oct 04 2022

A266196 Inverse permutation to A266195.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 13, 11, 15, 17, 22, 12, 14, 18, 20, 19, 26, 24, 31, 27, 33, 41, 43, 29, 45, 47, 49, 16, 21, 23, 35, 28, 39, 51, 75, 34, 57, 37, 77, 79, 83, 81, 89, 38, 53, 69, 91, 85, 99, 101, 105, 55, 107, 103, 109, 87, 111, 113, 115, 25, 30, 32, 61, 40, 59, 93, 97, 36, 54, 117, 166, 71, 186, 192, 194, 52, 121

Offset: 1

Antti Karttunen, Dec 26 2015

nonn

a(n) tells the time-step when n appears in "the match-making permutation" A266195.

Antti Karttunen, Table of n, a(n) for n = 1..631
Index entries for sequences that are permutations of the natural numbers

Inverse: A266195.

Cf. A266186, A266187.

A266197 Indices of "good matches" produced by match-making permutation A266195; numbers n for which A070939(A266195(n)) = A070939(A266195(n+1)), where A070939(n) gives the length of base-2 representation of n.

Original entry on oeis.org

2, 4, 5, 9, 10, 18, 19, 26, 34, 37, 38, 58, 61, 62, 66, 67, 129, 130, 133, 137, 138, 158, 286, 287, 290, 292, 301, 302, 311, 318, 365, 366, 371, 379, 382, 384, 561, 562, 563, 627, 628, 629, 630, 631, 633, 639, 644, 646, 680, 683, 688, 767, 768, 775, 776, 777, 778, 780, 913, 914, 915, 918, 919, 920, 923, 924, 925, 926, 927, 928, 930, 931, 932, 933, 939, 1315

Offset: 1

Antti Karttunen, Dec 26 2015

Numbers n for which A264982(n) = A264982(n+1).

It would be nice to know whether this sequence is infinite.

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 154 terms from Antti Karttunen)

Cf. A070939, A264982, A266195.

Cf. A265748, A265749 (give the first and second members of pairs indexed by this sequence).

A264982 Binary width of terms produced by match-making permutation: a(n) = A070939(A266195(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 6, 4, 5, 5, 5, 6, 4, 6, 5, 7, 5, 5, 6, 5, 7, 5, 7, 5, 6, 6, 7, 6, 6, 6, 7, 5, 8, 5, 8, 5, 8, 5, 8, 5, 9, 6, 7, 6, 7, 6, 8, 6, 7, 7, 8, 7, 7, 7, 8, 7, 8, 8, 8, 6, 8, 7, 8, 7, 9, 6, 9, 6, 9, 6, 9, 6, 10, 6, 9, 6, 9, 6, 10, 6, 10, 6, 9, 7, 8, 7, 9, 7, 10, 6, 10, 6, 10, 6, 11, 6, 11, 6, 11, 6, 11, 6, 11, 6, 11, 6, 12, 7

Offset: 1

Antti Karttunen, Dec 26 2015

Each n occurs A000079(n-1) = 2^(n-1) times in total.

Antti Karttunen, Table of n, a(n) for n = 1..2694

Cf. A000079, A070939, A266195.
Cf. A266186 (where n first appears, most likely also the positions of records).
Cf. A266187 (where n last appears).
Cf. A266197 (gives numbers n where a(n) = a(n+1)).

(define (A264982 n) (A070939 (A266195 n)))

a(n) = A070939(A266195(n)).

A265748 First members of "good matches" produced by match-making permutation: a(n) = A266195(A266197(n)).

Original entry on oeis.org

2, 4, 5, 9, 10, 18, 20, 21, 40, 42, 48, 96, 67, 84, 145, 134, 148, 193, 168, 290, 268, 336, 296, 386, 328, 592, 580, 536, 584, 645, 552, 771, 585, 772, 656, 1184, 1156, 1104, 1542, 1096, 1031, 1160, 1072, 1161, 2069, 1544, 1312, 2368, 1288, 1170, 1792, 1216, 1290, 2340, 2240, 2309, 3136, 4480, 2144, 2185, 3104, 2193, 2062, 2320, 2208, 2313, 2210

Offset: 1

Antti Karttunen, Dec 26 2015

Note that a number occurs both here and in A265749 if and only if it is a good match both with its predecessor and the successor in A266195.

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 154 terms from Antti Karttunen)

Cf. A070939, A266195, A266197.
Cf. A265749 (for the latter member).
Cf. also A235034, A235035.

(define (A265748 n) (A266195 (A266197 n)))

a(n) = A266195(A266197(n)).
Other identities. For all n >= 1:
A070939(a(n)) = A070939(A265749(n)). [By definition of "good match" in this context.]

A265749 Second members of "good matches" produced by match-making permutation: a(n) = A266195(1+A266197(n)).

Original entry on oeis.org

3, 5, 6, 10, 12, 20, 19, 24, 35, 48, 37, 69, 84, 97, 134, 160, 193, 164, 194, 268, 320, 385, 386, 297, 387, 770, 536, 641, 519, 608, 771, 648, 560, 594, 774, 1539, 1104, 1542, 1105, 1031, 1160, 1072, 1161, 1552, 2120, 1061, 1548, 3074, 1043, 1120, 1097, 1290, 1600, 2240, 2309, 3136, 2569, 4168, 2185, 3104, 2192, 2062, 2320, 2055, 2313, 2210, 3084

Offset: 1

Antti Karttunen, Dec 26 2015

nonn

Note that a number occurs both here and in A265748 if it is a good match both with its precedent and the successor in A266195.

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 154 terms from Antti Karttunen)

Cf. A070939, A266195, A266197.
Cf. A265748 (for the first member).
Cf. also A235034, A235035.

(define (A265749 n) (A266195 (+ 1 (A266197 n))))

a(n) = A266195(1+A266197(n)).
Other identities. For all n >= 1:
A070939(a(n)) = A070939(A265748(n)). [By definition of "good match" in this context.]

A266187 a(n) = A266196(A000225(n)); index of (2^n)-1 in A266195.

Original entry on oeis.org

1, 3, 8, 22, 49, 115, 258, 1040, 2610

Offset: 1

Antti Karttunen, Dec 26 2015

A264982(a(n)) is the last term k in A264982 for which A070939(k) = n.

Cf. A000225, A070939, A264982, A266195, A266196, A266186.

a(n) = A266196(A000225(n)).

A266194 Products of pairs made by match-making permutation: a(n) = A266195(n) * A266195(n+1).

Original entry on oeis.org

2, 6, 12, 20, 30, 48, 56, 63, 90, 120, 192, 176, 187, 221, 416, 448, 252, 360, 380, 627, 495, 510, 748, 1408, 1344, 504, 864, 1008, 1820, 1495, 1518, 1650, 1000, 1400, 2520, 3024, 2016, 1776, 2516, 1768, 3328, 3456, 3483, 3741, 3770, 3900, 3960, 4092, 7936, 9728, 3040, 3920, 3577, 4088, 7616, 5576, 3936, 6624, 9936, 9648, 5628, 8148, 13289

Offset: 1

Antti Karttunen, Dec 26 2015

Antti Karttunen, Table of n, a(n) for n = 1..2693

Cf. A048720, A057889, A266195, A266351.

a(n) = A266195(n) * A266195(n+1).
a(n) = A048720(A266195(n), A266195(n+1)).
a(n) = A057889(A057889(A266195(n)) * A057889(A266195(n+1))). [See the comment in A266351.]

A234741 a(n) is the base-2 carryless product of the prime factors of n; Encoding of the product of the polynomials over GF(2) represented by the prime factors of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 5, 10, 11, 12, 13, 14, 15, 16, 17, 10, 19, 20, 9, 22, 23, 24, 17, 26, 15, 28, 29, 30, 31, 32, 29, 34, 27, 20, 37, 38, 23, 40, 41, 18, 43, 44, 17, 46, 47, 48, 21, 34, 51, 52, 53, 30, 39, 56, 53, 58, 59, 60, 61, 62, 27, 64, 57, 58, 67

Offset: 1

Antti Karttunen, Jan 22 2014

nonn

"Encoding" means the number whose binary representation is given by the coefficients of the polynomial, e.g., 13=1101[2] encodes X^3+X^2+1. The product is the usual multiplication of polynomials in GF(2)[X] (or binary multiplication without carry-bits, cf. A048720).

a(n) <= n. [As all terms of the table A061858 are nonnegative]

			a(9) = a(3*3) = 5, as when we multiply 3 ('11' in binary) with itself, and discard the carry-bits, using XOR (A003987) instead of normal addition, we get:
   11
  110
-----
  101
that is, 5, as '101' is its binary representation. In other words, a(9) = a(3*3) = A048720(3,3) = 5.
Alternatively, 9 = 3*3, and 3=11[2] encodes the polynomial X+1, and (X+1)*(X+1) = X^2+1 in GF(2)[X], which is encoded as 101[2] = 5, therefore a(9) = 5. - _M. F. Hasler_, Feb 16 2014

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

A235034 gives the k for which a(k)=k.
A236833(n) gives the number of times n occurs in this sequence.
A236841 gives the same sequence sorted and duplicates removed, A236834 gives the numbers that do not occur here, A236835 gives numbers that occur more than once.
A325562(n) gives the number of iterations needed before one of the fixed points (terms of A235034) is reached.
Cf. also A048720, A061858, A234742, A236378, A091202/A091203, A235041/A235042, A266195, A325561, A325562.

A234741(n)={n=factor(n);n[,1]=apply(t->Pol(binary(t)),n[,1]);sum(i=1,#n=Vec(factorback(n))%2,n[i]<<(#n-i))} \\ M. F. Hasler, Feb 18 2014

a(0)=0, a(1)=1, and for n > 1, a(n) = A048720(A020639(n),a(n/A020639(n))). [A048720 used as a bivariate function]
Equally, for n with its unique prime factorization n = p_1 * ... * p_k, with the p_i not necessarily distinct primes, a(n) = p_1 x ... x p_k, where x stands for carryless multiplication defined in A048720, which is isomorphic to multiplication in GF(2)[X].
a(2n) = 2*a(n).
More generally, if A061858(x,y) = 0, then a(x*y) = a(x)*a(y).
a(A235034(n)) = A235034(n).
A236378(n) = n - a(n).

Term a(0) = 0 removed and a new primary definition added by Antti Karttunen, May 10 2019

A061858 Differences between the ordinary multiplication table A004247 and the carryless multiplication table for GF(2)[X] polynomials A048720, i.e., the effect of the carry bits in binary multiplication.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 12, 0, 8, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0

Offset: 0

Antti Karttunen, May 11 2001

			From _Peter Munn_, Jan 28 2021: (Start)
The top left 12 X 12 corner of the table:
      |  0   1   2   3   4   5   6   7   8   9  10  11
------+------------------------------------------------
   0  |  0   0   0   0   0   0   0   0   0   0   0   0
   1  |  0   0   0   0   0   0   0   0   0   0   0   0
   2  |  0   0   0   0   0   0   0   0   0   0   0   0
   3  |  0   0   0   4   0   0   8  12   0   0   0   4
   4  |  0   0   0   0   0   0   0   0   0   0   0   0
   5  |  0   0   0   0   0   8   0   8   0   0  16  16
   6  |  0   0   0   8   0   0  16  24   0   0   0   8
   7  |  0   0   0  12   0   8  24  28   0   0  16  28
   8  |  0   0   0   0   0   0   0   0   0   0   0   0
   9  |  0   0   0   0   0   0   0   0   0  16   0  16
  10  |  0   0   0   0   0  16   0  16   0   0  32  32
  11  |  0   0   0   4   0  16   8  28   0  16  32  52
(End)

"Zoomed in" variant: A061859.
Rows/columns 3, 5 and 7 are given by A048728, A048729, A048730.
Main diagonal divided by 4: A213673.
Numbers that generate no carries when multiplied in binary by 11_2: A003714, by 101_2: A048716, by 1001_2: A115845, by 10001_2: A115847, by 100001_2: A114086.
Other sequences related to the presence/absence of a carry in binary multiplication: A116361, A235034, A235040, A236378, A266195, A289726.

a(n) = A004247(n) - A048720(n).

cp's OEIS Frontend

A266186 a(n) = A266196(A000079(n)); indices of powers of 2 in A266195.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A266196 Inverse permutation to A266195.

Views

Author

Keywords

Comments

Links

Crossrefs

A266197 Indices of "good matches" produced by match-making permutation A266195; numbers n for which A070939(A266195(n)) = A070939(A266195(n+1)), where A070939(n) gives the length of base-2 representation of n.

Views

Author

Keywords

Comments

Links

Crossrefs

A264982 Binary width of terms produced by match-making permutation: a(n) = A070939(A266195(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A265748 First members of "good matches" produced by match-making permutation: a(n) = A266195(A266197(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A265749 Second members of "good matches" produced by match-making permutation: a(n) = A266195(1+A266197(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A266187 a(n) = A266196(A000225(n)); index of (2^n)-1 in A266195.

Views

Author

Keywords

Comments

Crossrefs

Formula

A266194 Products of pairs made by match-making permutation: a(n) = A266195(n) * A266195(n+1).

Views

Author

Keywords

Links

Crossrefs

Formula

A234741 a(n) is the base-2 carryless product of the prime factors of n; Encoding of the product of the polynomials over GF(2) represented by the prime factors of n (with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions