A277320 -id:A277320 - cp's OEIS frontend

A277699 Main diagonal of A277320: a(n) = A048720(n, A065621(n)).

1, 4, 9, 16, 57, 36, 49, 64, 209, 228, 217, 144, 233, 196, 225, 256, 801, 836, 809, 912, 793, 868, 785, 576, 1009, 932, 1017, 784, 969, 900, 961, 1024, 3137, 3204, 3145, 3344, 3193, 3236, 3185, 3648, 3217, 3172, 3225, 3472, 3241, 3140, 3233, 2304, 3937, 4036, 3945, 3728, 3929, 4068, 3921

Offset: 1

Antti Karttunen, Nov 01 2016

Cf. A000290, A023758, A048720, A065621, A277320.
Cf. A277704, A277706 (the positions of squares/nonsquares in this sequence).
Cf. A277805 (nonsquares in the order of appearance).

(define (A277699 n) (A277320bi n n)) ;; Code for A277320bi given in A277320.

a(n) = A277320(n,n) = A048720(n, A065621(n)).

For n > 1, a(A023758(n)) = A000290(A023758(n)).

A277199 Transpose of square array A277320.

Original entry on oeis.org

1, 2, 2, 7, 4, 3, 4, 14, 6, 4, 13, 8, 9, 8, 5, 14, 26, 12, 28, 10, 6, 11, 28, 23, 16, 27, 12, 7, 8, 22, 18, 52, 20, 18, 14, 8, 25, 16, 29, 56, 57, 24, 21, 16, 9, 26, 50, 24, 44, 54, 46, 28, 56, 18, 10, 31, 52, 43, 32, 39, 36, 35, 32, 63, 20, 11, 28, 62, 46, 100, 40, 58, 42, 104, 36, 54, 22, 12, 21, 56, 33, 104, 125, 48, 49, 112, 101, 40, 49, 24, 13

Offset: 1

Antti Karttunen, Nov 01 2016

See A277320.

			The top left corner of the array:
   1,  2,  7,  4,  13,  14,  11,   8,  25,  26,  31,  28
   2,  4, 14,  8,  26,  28,  22,  16,  50,  52,  62,  56
   3,  6,  9, 12,  23,  18,  29,  24,  43,  46,  33,  36
   4,  8, 28, 16,  52,  56,  44,  32, 100, 104, 124, 112
   5, 10, 27, 20,  57,  54,  39,  40, 125, 114,  99, 108
   6, 12, 18, 24,  46,  36,  58,  48,  86,  92,  66,  72
   7, 14, 21, 28,  35,  42,  49,  56,  79,  70,  93,  84
   8, 16, 56, 32, 104, 112,  88,  64, 200, 208, 248, 224
   9, 18, 63, 36, 101, 126,  83,  72, 209, 202, 231, 252
  10, 20, 54, 40, 114, 108,  78,  80, 250, 228, 198, 216
  11, 22, 49, 44, 127,  98,  69,  88, 227, 254, 217, 196
  12, 24, 36, 48,  92,  72, 116,  96, 172, 184, 132, 144

Transpose: A277320.
Main diagonal: A277699.
Row 1: A065621, Row 3: 3: A277823.
Column 1: A000027, Column 3: A048727.
Cf. A048720.

(define (A277199 n) (A277320bi (A004736 n) (A002260 n))) ;; Code for A277320bi given in A277320.

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

A048720 Multiplication table {0..i} X {0..j} of binary polynomials (polynomials over GF(2)) interpreted as binary vectors, then written in base 10; or, binary multiplication without carries.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 5, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 10, 20, 20, 10, 14, 8, 0, 0, 9, 16, 9, 24, 17, 24, 9, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 27, 20, 27, 32, 27, 20, 11, 0

Offset: 0

Antti Karttunen, Apr 26 1999

Essentially same as A091257 but computed starting from offset 0 instead of 1.
Each polynomial in GF(2)[X] is encoded as the number whose binary representation is given by the coefficients of the polynomial, e.g., 13 = 2^3 + 2^2 + 2^0 = 1101_2 encodes 1*X^3 + 1*X^2 + 0*X^1 + 1*X^0 = X^3 + X^2 + X^0. - Antti Karttunen and Peter Munn, Jan 22 2021
To listen to this sequence, I find instrument 99 (crystal) works well with the other parameters defaulted. - Peter Munn, Nov 01 2022

			Top left corner of array:
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ...
  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  0  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 ...
  0  3  6  5 12 15 10  9 24 27 30 29 20 23 18 17 ...
  ...
From _Antti Karttunen_ and _Peter Munn_, Jan 23 2021: (Start)
Multiplying 10 (= 1010_2) and 11 (= 1011_2), in binary results in:
     1011
  *  1010
  -------
   c1011
  1011
  -------
  1101110  (110 in decimal),
and we see that there is a carry-bit (marked c) affecting the result.
In carryless binary multiplication, the second part of the process (in which the intermediate results are summed) looks like this:
    1011
  1011
  -------
  1001110  (78 in decimal).
(End)

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Antti Karttunen, Scheme-program for computing this sequence.
N. J. A. Sloane, Transforms: Maple implementation of binary eXclusive OR (XORnos).
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A051776 (Nim-product), A091257 (subtable).
Carryless multiplication in other bases: A325820 (3), A059692 (10).
Ordinary {0..i} * {0..j} multiplication table: A004247 and its differences from this: A061858 (which lists further sequences related to presence/absence of carry in binary multiplication).
Carryless product of the prime factors of n: A234741.
Binary irreducible polynomials ("X-primes"): A014580, factorization table: A256170, table of "X-powers": A048723, powers of 3: A001317, rearranged subtable with distinct terms (comparable to A054582): A277820.
See A014580 for further sequences related to the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the integer encoding.
Row/column 3: A048724 (even bisection of A003188), 5: A048725, 6: A048726, 7: A048727; main diagonal: A000695.
Associated additive operation: A003987.
Equivalent sequences, as compared with standard integer multiplication: A048631 (factorials), A091242 (composites), A091255 (gcd), A091256 (lcm), A280500 (division).
See A091202 (and its variants) and A278233 for maps from/to ordinary multiplication.
See A115871, A115872 and A277320 for tables related to cross-domain congruences.

trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of the triangular numbers
# Binary multiplication of nn and mm, but without carries (use XOR instead of ADD):
Xmult := proc(nn,mm) local n,m,s; n := nn; m := mm; s := 0; while (n > 0) do if(1 = (n mod 2)) then s := XORnos(s,m); fi; n := floor(n/2); # Shift n right one bit. m := m*2; # Shift m left one bit. od; RETURN(s); end;

trinv[n_] := Floor[(1 + Sqrt[1 + 8*n])/2];
Xmult[nn_, mm_] := Module[{n = nn, m = mm, s = 0}, While[n > 0, If[1 == Mod[n, 2], s = BitXor[s, m]]; n = Floor[n/2]; m = m*2]; Return[s]];
a[n_] := Xmult[(trinv[n] - 1)*((1/2)*trinv[n] + 1) - n, n - (trinv[n]*(trinv[n] - 1))/2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 16 2015, updated Mar 06 2016 after Maple *)

up_to = 104;
A048720sq(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A048720list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A048720sq(col, a-col))); (v); };
v048720 = A048720list(up_to);
A048720(n) = v048720[1+n]; \\ Antti Karttunen, Feb 15 2021

a(n) = Xmult( (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)) );
T(2b, c)=T(c, 2b)=T(b, 2c)=2T(b, c); T(2b+1, c)=T(c, 2b+1)=2T(b, c) XOR c - Henry Bottomley, Mar 16 2001
For n >= 0, A003188(2n) = T(n, 3); A003188(2n+1) = T(n, 3) XOR 1, where XOR is the bitwise exclusive-or operator, A003987. - Peter Munn, Feb 11 2021

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

A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), 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

1, 3, 2, 5, 6, 7, 15, 10, 9, 4, 17, 30, 27, 12, 13, 51, 34, 45, 20, 23, 14, 85, 102, 119, 60, 57, 18, 11, 255, 170, 153, 68, 75, 54, 29, 8, 257, 510, 427, 204, 221, 90, 39, 24, 25, 771, 514, 765, 340, 359, 238, 105, 40, 43, 26, 1285, 1542, 1799, 1020, 937, 306, 187, 120, 125, 46, 31, 3855, 2570, 2313, 1028, 1275, 854, 461, 136, 135, 114, 33, 28

Offset: 1

Antti Karttunen, Nov 01 2016

For all n >= 1, A277818 (= A268389(n)+1) gives the (one-based) index of the column where n is located in this array, while A268671(n) gives the (one-based) index of the row where it is on.

This array is obtained when one selects from A277320 the columns 1, 3, 5, 15, 17, 51, ..., i.e., those with an index A001317(k).

			The top left corner of the array:
   1,  3,   5,  15,  17,   51,   85,  255,   257,   771,  1285,  3855
   2,  6,  10,  30,  34,  102,  170,  510,   514,  1542,  2570,  7710
   7,  9,  27,  45, 119,  153,  427,  765,  1799,  2313,  6939, 11565
   4, 12,  20,  60,  68,  204,  340, 1020,  1028,  3084,  5140, 15420
  13, 23,  57,  75, 221,  359,  937, 1275,  3341,  5911, 14649, 19275
  14, 18,  54,  90, 238,  306,  854, 1530,  3598,  4626, 13878, 23130
  11, 29,  39, 105, 187,  461,  599, 1785,  2827,  7453, 10023, 26985
   8, 24,  40, 120, 136,  408,  680, 2040,  2056,  6168, 10280, 30840
  25, 43, 125, 135, 393,  667, 1965, 2295,  6425, 11051, 32125, 34695
  26, 46, 114, 150, 442,  718, 1874, 2550,  6682, 11822, 29298, 38550
  31, 33,  99, 165, 495,  561, 1619, 2805,  7967,  8481, 25443, 42405
  28, 36, 108, 180, 476,  612, 1708, 3060,  7196,  9252, 27756, 46260
  21, 63,  65, 195, 325,  975, 1105, 3315,  5397, 16191, 16705, 50115
  22, 58,  78, 210, 374,  922, 1198, 3570,  5654, 14906, 20046, 53970
  19, 53,  95, 225, 291,  869, 1455, 3825,  4883, 13621, 24415, 57825
  16, 48,  80, 240, 272,  816, 1360, 4080,  4112, 12336, 20560, 61680
  49, 83, 245, 287, 801, 1379, 4005, 4335, 12593, 21331, 62965, 73247
  50, 86, 250, 270, 786, 1334, 3930, 4590, 12850, 22102, 64250, 69390
  55, 89, 235, 317, 839, 1481, 3675, 4845, 14135, 22873, 60395, 80957

Inverse permutation: A277821.
Transpose: A277819.
Cf. A048724, A065621.
Row 1: A001317.
Column 1: A065621, column 2: A277823, column 3: A277825.
Cf. A268389, A277818, A268671.
Cf. also A193231, A277320, A277810, A276618 (A099884), A248513.
Other related tables or permutations: A277880, A277901.

(define (A277820 n) (A277820bi (A002260 n) (A004736 n)))
(define (A277820bi row col) (if (= 1 col) (A065621 row) (A048724 (A277820bi row (- col 1)))))

A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)).
A(r,c) = A048675(A277810(r,c)).
As a composition of other permutations:
a(n) = A277901(A277880(n)).

A277823 a(n) = A048724(A065621(n)).

Original entry on oeis.org

3, 6, 9, 12, 23, 18, 29, 24, 43, 46, 33, 36, 63, 58, 53, 48, 83, 86, 89, 92, 71, 66, 77, 72, 123, 126, 113, 116, 111, 106, 101, 96, 163, 166, 169, 172, 183, 178, 189, 184, 139, 142, 129, 132, 159, 154, 149, 144, 243, 246, 249, 252, 231, 226, 237, 232, 219, 222, 209, 212, 207, 202, 197, 192, 323, 326, 329, 332, 343, 338, 349, 344, 363, 366, 353

Offset: 1

Antti Karttunen, Nov 01 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A048720, A065621.
Column 2 of A277820.
Column 3 of A277320.

def A277823(n): return (m:=n^ (n&~-n)<<1)^m<<1 # Chai Wah Wu, Jun 29 2022

(define (A277823 n) (A048724 (A065621 n)))
(define (A277823 n) (A048720bi (A065621 n) 3))
(define (A277823 n) (A277320bi n 3))

a(n) = A048724(A065621(n)).

a(n) = A277320(n,3) = A048720(A065621(n),3).

A379121 Odd squares k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

Original entry on oeis.org

225, 3025, 3249, 12321, 29241, 38025, 91809, 216225, 247009, 354025, 408321, 751689, 772641, 855625, 919681, 1366561, 1595169, 3814209, 9828225, 11189025, 12173121, 12709225, 29430625, 47927929, 52403121, 66471409, 67486225, 77457601, 80263681, 94148209, 100661089, 110397049, 126540001, 204232681, 264875625, 328878225

Offset: 1

Antti Karttunen, Dec 18 2024

nonn

Of the first 2025 terms, only two, a(520) and a(1087) have multiple solutions. See the examples.

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.

A277825 a(n) = A048725(A065621(n)) = A048720(A065621(n),5).

Original entry on oeis.org

5, 10, 27, 20, 57, 54, 39, 40, 125, 114, 99, 108, 65, 78, 95, 80, 245, 250, 235, 228, 201, 198, 215, 216, 141, 130, 147, 156, 177, 190, 175, 160, 485, 490, 507, 500, 473, 470, 455, 456, 413, 402, 387, 396, 417, 430, 447, 432, 277, 282, 267, 260, 297, 294, 311, 312, 365, 354, 371, 380, 337, 350, 335, 320, 965, 970, 987, 980, 1017, 1014, 999, 1000

Offset: 1

Antti Karttunen, Nov 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Column 3 of A277820, Column 5 of A277320.

Cf. A048720, A048724, A048725, A065621, A277823.

def A277825(n): return (m:=n^ (n&~-n)<<1)^m<<2 # Chai Wah Wu, Jun 29 2022

(define (A277825 n) (A048724 (A277823 n)))
(define (A277825 n) (A048720bi (A065621 n) 5)) ;; For A048720bi see the Scheme-program attachment in A048720.

a(n) = A048724(A277823(n)) = A048725(A065621(n)).

a(n) = A048720(A065621(n),5).

A278239 a(n) = A278233(A277699(n)).

Original entry on oeis.org

1, 4, 6, 16, 12, 60, 6, 64, 30, 180, 6, 240, 12, 60, 24, 256, 240, 420, 6, 720, 12, 60, 30, 960, 6, 180, 60, 240, 30, 360, 30, 1024, 30, 5040, 30, 1680, 6, 60, 360, 2880, 30, 180, 210, 240, 120, 420, 6, 3840, 60, 60, 96, 720, 210, 1260, 6, 960, 60, 420, 30, 2160, 12, 420, 60, 4096, 180, 420, 30, 45360, 60, 420, 30, 6720, 30, 60, 840, 240, 30, 12600, 30, 11520, 12

Offset: 1

Antti Karttunen, Nov 17 2016

nonn

Prime factorization sentinel computed for A277699. Factorization is essentially done in the polynomial ring GF(2)[X].

Cf. A048720, A065621, A277320, A277699, A278233, A278238.

(define (A278239 n) (A278233 (A277699 n)))

a(n) = A278233(A277699(n)).

cp's OEIS Frontend

A277699 Main diagonal of A277320: a(n) = A048720(n, A065621(n)).

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A277199 Transpose of square array A277320.

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A048720 Multiplication table {0..i} X {0..j} of binary polynomials (polynomials over GF(2)) interpreted as binary vectors, then written in base 10; or, binary multiplication without carries.

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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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A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), 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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A277823 a(n) = A048724(A065621(n)).

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A379121 Odd squares k for which A379113(k) > 1, i.e., k that have a proper unitary divisor d > 1 such that A048720(A065621(sigma(d)),sigma(k/d)) is equal to sigma(k).

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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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