A115872 -id:A115872 - cp's OEIS frontend

A048717 Binary expansion matches ((0)00(1)11)(0).

0, 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, 63, 96, 99, 102, 103, 112, 115, 120, 124, 126, 127, 192, 195, 198, 199, 204, 206, 207, 224, 227, 230, 231, 240, 243, 248, 252, 254, 255, 384, 387, 390, 391

Offset: 0

Antti Karttunen, Mar 30 1999

In binary expansion, 1-bits occur only in groups of two or more, separated from other such groups by at least two 0-bits.

Integers that satisfy A048727(n) = 3*n.

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

Row 3 of A115872. Superset of A048719. Cf. A048733.

filterQ[n_] := !MatchQ[IntegerDigits[n, 2], {1}|{1, 0, _}|{_, 0, 1}|{_, 1, 0, 1, _}|{_, 0, 1, 0, _}];
Select[Range[0, 400], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

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.

A115772 Integers i such that 13*i = A048720bi(21,i).

Original entry on oeis.org

0, 5, 10, 15, 20, 21, 30, 31, 40, 42, 45, 47, 60, 61, 62, 63, 80, 84, 85, 90, 94, 95, 120, 122, 124, 125, 126, 127, 160, 165, 168, 170, 173, 175, 180, 181, 188, 189, 190, 191, 240, 244, 245, 248, 250, 252, 253, 254, 255, 320, 330, 336, 340, 341, 346, 350, 351

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

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

a(n) appears to be the set of all n that can be expressed as x OR 4x for the bitwise OR operation. [From Gary Detlefs Dec 20 2010]

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

Row 13 of A115872. Cf. A048717, A115767, A115770. Superset of A115774 ? A115776 gives the terms which are not in A115774. A115773 shows this sequence in binary.

Cf. A178891

A284270 Square array A(r,c) = A048720(A065621(r), c) mod r, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 3, 4, 4, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 2, 1, 0, 7, 0, 0, 2, 0, 1, 0, 2, 0, 5, 6, 0, 0, 0, 0, 0, 0, 4, 0, 7, 2, 9, 0, 0, 0, 0, 4, 0, 2, 0, 1, 6, 7, 4, 0, 0, 1, 0, 1, 4, 0, 0, 8, 4, 0, 8, 8, 0, 0, 0, 0, 4, 0, 4, 0, 5, 4, 3, 0, 3, 8, 0, 0, 2, 0, 2, 0, 6, 0, 7, 2, 0, 4, 11, 2, 4

Offset: 1

Antti Karttunen, Apr 10 2017

			The top left 17 x 19 corner of the array:
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   1,  2,  0,  1,  0,  0,  0,  2,  0,  0,  1,  0,  2,  0,  0,  1,  2
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   3,  1,  3,  2,  2,  1,  0,  4,  1,  4,  2,  2,  1,  0,  0,  3,  1
   2,  4,  0,  2,  0,  0,  0,  4,  0,  0,  2,  0,  4,  0,  0,  2,  4
   4,  1,  1,  2,  4,  2,  0,  4,  6,  1,  6,  4,  1,  0,  0,  1,  5
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
   7,  5,  7,  1,  8,  5,  7,  2,  2,  7,  2,  1,  1,  5,  0,  4,  6
   6,  2,  6,  4,  4,  2,  0,  8,  2,  8,  4,  4,  2,  0,  0,  6,  2
   9,  7,  0,  3,  0,  0,  5,  6,  0,  0,  8,  0,  1, 10,  0,  1,  0
   4,  8,  0,  4,  0,  0,  0,  8,  0,  0,  4,  0,  8,  0,  0,  4,  8
   8,  3, 11,  6,  0,  9,  3, 12,  7,  0,  8,  5, 12,  6,  0, 11,  0
   8,  2,  2,  4,  8,  4,  0,  8, 12,  2, 12,  8,  2,  0,  0,  2, 10
   4,  8,  8,  1,  5,  1,  1,  2,  4, 10,  8,  2,  4,  2,  0,  4,  6
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
  15, 13, 15,  9,  7, 13, 15,  1, 16, 14, 16,  9,  7, 13, 15,  2,  2
  14, 10, 14,  2, 16, 10, 14,  4,  4, 14,  4,  2,  2, 10,  0,  8, 12
  17, 15, 13, 11,  7,  7,  0,  3,  0, 14,  6, 14, 16,  0, 13,  6,  3

Cf. A048720, A065621, A115872, A277320, A284269 (transpose), A284273 (main diagonal), A284552 (column 1).

Row 3: A284557.

(define (A284270 n) (A284270bi (A002260 n) (A004736 n)))
(define (A284270bi row col) (modulo (A277320bi row col) row)) ;; For A277320bi, see in A277320.

A(r,c) = A277320(r,c) mod r = A048720(A065621(r), c) mod r.

A325570 Numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 131, 137, 139, 141, 143, 145, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 175, 177, 179, 181

Offset: 1

Antti Karttunen, May 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048720, A065621, A115872, A115873.
Positions of ones in A325565 and A325566.
Cf. A065091 (a subsequence), A325571 (the composite terms), A325572 (complement).
Subsequence of A005408 (odd numbers).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
isA325570(n) = fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(d==n)));

A115770 Integers i such that 7*i = A048720bi(11, i), where A048720bi implements the dyadic function given in A048720 (see A001317).

Original entry on oeis.org

0, 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, 224, 240, 248, 252, 254, 255, 448, 455, 480, 496, 504, 508, 510, 511, 896, 903, 910, 911, 960, 967, 992, 1008, 1016, 1020, 1022, 1023, 1792, 1799, 1806, 1807, 1820, 1822, 1823, 1920

Offset: 0

Antti Karttunen, Jan 30 2006

nonn

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

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

Row 7 of A115872 (conjecture: also row 5).
Cf. A001317, A048717, A048720, A115767, A115772, A115774, A115776.
A115771 shows this sequence in binary.

A325568 a(1) = 0; for n > 1, a(n) = 1 + a(A325567(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 1, 4, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 5, 2, 2, 2, 4, 1, 2, 1, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 2, 3, 1, 3, 1, 3, 1, 5, 1, 2, 2, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 3, 2, 2, 1, 6, 1, 3, 1, 3, 1, 2, 1, 4, 2

Offset: 1

Antti Karttunen, May 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001222, A048720, A065621, A115872, A325565, A325566, A325567, A325569.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A325567(n) = if(1==n,n,fordiv(n,d,if((d>1)&&A048720(A065621(n/d),d)==n,return(n/d))));
A325568(n) = if(1==n,0,1+A325568(A325567(n)));

a(1) = 0; for n > 1, a(n) = 1 + a(A325567(n)).

a(n) <= A001222(n) for all n.

A340351 Square array, read by descending antidiagonals, where row n gives all solutions k > 0 to A000120(k)=A000120(k*n), A000120 is the Hamming weight.

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, 27, 7, 14, 1, 9, 8, 24, 6, 28, 12, 15, 2, 15, 10, 9, 28, 7, 30, 14, 19, 3, 30, 7, 11, 10, 30, 8, 31, 15, 28, 4, 31, 14, 3, 12, 11, 31, 9, 39, 24, 30, 5, 43, 15, 6, 3, 13, 12

Offset: 1

Thomas Scheuerle, Jan 05 2021

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.
Values are different from those in A115872, because we use multiplication with carry here.

			Eight initial terms of rows 1 - 8 are listed below:
   1:  1,  2,  3,   4,   5,   6,   7,   8, ...
   2:  1,  2,  3,   4,   5,   6,   7,   8, ...
   3:  3,  6,  7,  12,  14,  15,  24,  28, ...
   4:  1,  2,  3,   4,   5,   6,   7,   8, ...
   5:  7, 14, 15,  27,  28,  30,  31,  39, ...
   6:  3,  6,  7,  12,  14,  15,  24,  28, ...
   7:  7, 14, 15,  19,  28,  30,  31,  37, ...
   8:  1,  2,  3,   4,   5,   6,   7,   8, ...
a(6,3) = 7 because: 7 in binary is 111 and 6*7 = 42 in binary is 101001, both have 3 bits set to 1.

Cf. A000120, A292849 (1st column), A340069, A077459 (3rd row).

function [a] = A340351(max_n)
    for n = 1:max_n
        m = 1;
        k = 1;
        while m < max_n
            c = length(find(bitget(k,1:32)== 1));
            if c == length(find(bitget(n*k,1:32)== 1))
                a(n,m) = k;
                m = m+1;
            end
            k = k +1;
        end
    end
end

A114386 Integers i such that 57*i = 73 X i.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 73, 90, 91, 108, 109, 126, 127, 144, 146, 153, 155, 180, 182, 189, 191, 216, 217, 218, 219, 252, 253, 254, 255, 288, 292, 297, 301, 306, 310, 315, 319, 360, 361, 364, 365, 378, 379, 382, 383, 432, 434, 436, 438, 441, 443

Offset: 1

Antti Karttunen, Feb 07 2006

nonn

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

Row 57 of A115872. A114387 shows this sequence in binary.

g:= proc(n) local L,i;
  L:= op(convert(n,base,2));
  L:= [0$6,L] + [0$3,L,0$3] + [L,0$6] mod 2;
  add(L[i]*2^(i-1),i=1..nops(L));
end proc:
select(t -> 57*t = g(t), [$0..1000]); # Robert Israel, Jan 31 2021

Offset corrected by Robert Israel, Jan 31 2021

cp's OEIS Frontend

A048717 Binary expansion matches ((0)*00(1*)11)*(0*).

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Mathematica

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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A115772 Integers i such that 13*i = A048720bi(21,i).

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A284270 Square array A(r,c) = A048720(A065621(r), c) mod r, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A325570 Numbers n that have no divisor d > 1 such that A048720(A065621(d),n/d) = n.

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PARI

A115770 Integers i such that 7*i = A048720bi(11, i), where A048720bi implements the dyadic function given in A048720 (see A001317).

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A325568 a(1) = 0; for n > 1, a(n) = 1 + a(A325567(n)).

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A340351 Square array, read by descending antidiagonals, where row n gives all solutions k > 0 to A000120(k)=A000120(k*n), A000120 is the Hamming weight.

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MATLAB

A114386 Integers i such that 57*i = 73 X i.

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A048717 Binary expansion matches ((0)00(1)11)(0).