A065621 -id:A065621 - cp's OEIS frontend

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.

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

A277320 Square array A(r,c) = A048720(A065621(r), c), 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, 2, 2, 3, 4, 7, 4, 6, 14, 4, 5, 8, 9, 8, 13, 6, 10, 28, 12, 26, 14, 7, 12, 27, 16, 23, 28, 11, 8, 14, 18, 20, 52, 18, 22, 8, 9, 16, 21, 24, 57, 56, 29, 16, 25, 10, 18, 56, 28, 46, 54, 44, 24, 50, 26, 11, 20, 63, 32, 35, 36, 39, 32, 43, 52, 31, 12, 22, 54, 36, 104, 42, 58, 40, 100, 46, 62, 28, 13, 24, 49, 40, 101, 112, 49, 48, 125, 104, 33, 56, 21

Offset: 1

Antti Karttunen, Nov 01 2016

			The top left corner of the array:
   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12
   2,   4,   6,   8,  10,  12,  14,  16,  18,  20,  22,  24
   7,  14,   9,  28,  27,  18,  21,  56,  63,  54,  49,  36
   4,   8,  12,  16,  20,  24,  28,  32,  36,  40,  44,  48
  13,  26,  23,  52,  57,  46,  35, 104, 101, 114, 127,  92
  14,  28,  18,  56,  54,  36,  42, 112, 126, 108,  98,  72
  11,  22,  29,  44,  39,  58,  49,  88,  83,  78,  69, 116
   8,  16,  24,  32,  40,  48,  56,  64,  72,  80,  88,  96
  25,  50,  43, 100, 125,  86,  79, 200, 209, 250, 227, 172
  26,  52,  46, 104, 114,  92,  70, 208, 202, 228, 254, 184
  31,  62,  33, 124,  99,  66,  93, 248, 231, 198, 217, 132
  28,  56,  36, 112, 108,  72,  84, 224, 252, 216, 196, 144
  21,  42,  63,  84,  65, 126, 107, 168, 189, 130, 151, 252
  22,  44,  58,  88,  78, 116,  98, 176, 166, 156, 138, 232
  19,  38,  53,  76,  95, 106, 121, 152, 139, 190, 173, 212
  16,  32,  48,  64,  80,  96, 112, 128, 144, 160, 176, 192
  49,  98,  83, 196, 245, 166, 151, 392, 441, 490, 475, 332
  50, 100,  86, 200, 250, 172, 158, 400, 418, 500, 454, 344
  55, 110,  89, 220, 235, 178, 133, 440, 399, 470, 481, 356

Transpose: A277199.
Main diagonal: A277699.
Row 1: A000027, Row 3: A048727.
Column 1: A065621, Column 3: A277823, Column 5: A277825.
Cf. A277820 (array obtained by selecting only the columns with an index A001317(k), k=0..).
Cf. A048720, A115872.

(define (A277320 n) (A277320bi (A002260 n) (A004736 n)))
(define (A277320bi row col) (A048720bi (A065621 row) col))

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

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

Original entry on oeis.org

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)).

A325567 a(1) = 1; for n > 1, a(n) is the largest proper divisor d of n such that A048720(A065621(d),n/d) is equal to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 11, 2, 5, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 16, 7, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 13, 22, 1, 4, 1, 10, 1, 24, 1, 2, 5, 4, 1, 2, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 32, 1, 14, 1, 4, 1, 2, 1, 8, 7

Offset: 1

Antti Karttunen, May 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A032742, A048720, A065621, A115872, A325565, A325566, A325568.

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))));

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)).

A325565 a(n) is the number of such divisors d of n that A048720(A065621(d),n/d) is equal to n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 1, 4, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 6, 2, 2, 2, 6, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 7, 2, 4, 1, 3, 1, 4, 1, 8, 1, 2, 2, 3, 1, 2, 1, 5, 1, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 6, 1, 4, 1, 3, 1, 2, 1, 4, 2

Offset: 1

Antti Karttunen, May 09 2019

nonn

Equally, a(n) is number of such pairs of natural numbers t, u that A048720(t,u) = n and A065620(t)*u = n.

Cf. A000005, A000265, A001511, A048720, A065620, A065621, A091220, A115872, A325566, A325567, A325568, A325569, A325570 (positions of ones).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A325565(n) = sumdiv(n,d,A048720(A065621(d),n/d)==n);

A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<A065620
A325565(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sum(d=1,n,my(q = Pol(binary(d))*Mod(1, 2)); (0==(p%q) && (n==(A065620(d)*fromdigits(Vec(lift(p/q)),2))))); };

a(n) = Sum_{d|n} [A048720(A065621(d),n/d) == n], where [ ] is the Iverson bracket.
a(n) / a(A000265(n)) = A001511(n).
a(n) <= A000005(n) for all n.
a(n) <= A091220(n) for all n.

A379113 a(1) = 1; for n > 1, a(n) is the greatest proper unitary divisor d of n such that A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 3, 1, 1, 1, 1, 5, 7, 11, 1, 3, 1, 2, 1, 7, 1, 15, 1, 1, 3, 1, 7, 1, 1, 1, 3, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 2, 3, 13, 1, 1, 11, 7, 3, 2, 1, 15, 1, 31, 7, 1, 5, 33, 1, 1, 3, 35, 1, 9, 1, 1, 3, 19, 7, 6, 1, 5, 1, 1, 1, 21, 1, 43, 3, 11, 1, 1, 7, 23, 31, 47, 1, 3, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Dec 17 2024

nonn

Cf. A000203, A048720, A065621, A379114 (positions of terms > 1), A379119.

Cf. also A325567.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A379113(n) = if(1==n,n,my(s=sigma(n)); fordiv(n,d,if((d>1) && 1==gcd(d,n/d) && A048720(A065621(sigma(n/d)),sigma(d))==s,return(n/d))));

a(n) = n/A379119(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).

A325566 a(n) is the largest divisor d of n such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 1, 16, 1, 6, 1, 4, 3, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 32, 11, 2, 5, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 16, 7, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 64, 13, 22, 1, 4, 1, 10, 1, 24, 1, 2, 5, 4, 1, 2, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 32, 1, 14, 1, 4, 1, 2, 1, 8, 7

Offset: 1

Antti Karttunen, May 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A048720, A065621, A325565, A325567, A325570 (positions of ones).

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A325566(n) = fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(n/d)));

A246163 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A065621(1+a(n)), a(A091242(n)) = A048724(a(n)), where A065621(n) and A048724(n) give the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 7, 3, 6, 9, 8, 5, 10, 27, 4, 24, 11, 15, 30, 45, 12, 40, 26, 29, 17, 34, 119, 20, 25, 120, 46, 39, 51, 102, 14, 153, 60, 43, 136, 114, 31, 105, 85, 170, 44, 18, 427, 68, 125, 408, 13, 150, 33, 187, 255, 510, 116, 54, 41, 765, 204, 135, 28, 680, 16, 23, 442, 99, 461, 257, 35, 514, 156, 90, 123, 1799, 118, 340, 393, 36

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)) and A065621/A048724, the latter which themselves are permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246161.
Because 3 is the only evil number in A014580, it implies that, apart from a(3)=7, odious numbers occur in odious positions only (along with many evil numbers that also occur in odious positions).
Furthermore, all terms of A246158 reside in infinite cycles, and apart from 4 and 8, all of them reside in separate cycles. The infinite cycle containing 4 and 8 goes as: ..., 2091, 97, 47, 13, 11, 4, 3, 7, 8, 5, 6, 9, 10, 27, 46, 408, 2535, ... and it is only because a(3) = 7, that it can temporarily switch back from evil terms to odious terms, until right after a(8) = 5 it is finally doomed to the eternal evilness.
Please see also the comments at A246201 and A246161.

Inverse: A246164.
Similar or related permutations: A246205, A193231, A246201, A234026, A245701, A234612, A246161, A246203.
Cf. A010060, A014580, A048724, A065621, A091225, A091226, A091245.

a(1) = 1, and for n > 1, if n is in A014580, a(n) = A065621(1+a(A091226(n))), otherwise a(n) = A048724(a(A091245(n))).
As a composition of related permutations:
a(n) = A193231(A246201(n)).
a(n) = A234026(A245701(n)).
a(n) = A234612(A246161(n)).
a(n) = A193231(A246203(A193231(n))).
Other identities:
For all n > 1, A010060(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (A014580) to odious numbers and the corresponding representations of reducible polynomials (A091242) to evil numbers, in some order].

cp's OEIS Frontend

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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A277320 Square array A(r,c) = A048720(A065621(r), c), 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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A277699 Main diagonal of A277320: a(n) = A048720(n, A065621(n)).

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A325567 a(1) = 1; for n > 1, a(n) is the largest proper divisor d of n such that A048720(A065621(d),n/d) is equal to n.

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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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A325565 a(n) is the number of such divisors d of n that A048720(A065621(d),n/d) is equal to n.

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A379113 a(1) = 1; for n > 1, a(n) is the greatest proper unitary divisor d of n such that A048720(A065621(sigma(d)),sigma(n/d)) is equal to sigma(n).

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

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