A341520 -id:A341520 - cp's OEIS frontend

A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 18, 20, 20, 18, 14, 8, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12

Offset: 1

Marc LeBrun

Or, triangle X(n,m) = T(n-m+1,m) read by rows, in which row n gives the numbers n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n.
Radius of incircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 16 2001
A permutation of A061017. - Matthew Vandermast, Feb 28 2003
In the proof of countability of rational numbers they are arranged in a square array. a(n) = p*q where p/q is the corresponding rational number as read from the array. - Amarnath Murthy, May 29 2003
Permanent of upper right n X n corner is A000442. - Marc LeBrun, Dec 11 2003
Row 12 gives total number of partridges, turtle doves, ... and drummers drumming that you have received at the end of the Twelve Days of Christmas song. - Alonso del Arte, Jun 17 2005
Consider a particle with spin S (a half-integer) and 2S+1 quantum states |m>, m = -S,-S+1,...,S-1,S. Then the matrix element  = sqrt((S+m+1)(S-m)) of the spin-raising operator is the square-root of the triangular (tabl) element T(r,o) of this sequence in row r = 2S, and at offset o=2(S+m). T(r,o) is also the intensity || of the transition between the states |m> and |m+1>. For example, the five transitions between the 6 states of a spin S=5/2 particle have relative intensities 5,8,9,8,5. The total intensity of all spin 5/2 transitions (relative to spin 1/2) is 35, which is the tetrahedral number A000292(5). - Stanislav Sykora, May 26 2012
Sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k) = n. See A098359. - Charlie Marion, Apr 22 2013
T(n, k) is also the (k-1)-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n. - Stefano Spezia, Jul 12 2019
From Eric Lengyel, Jun 28 2023: (Start)
X(n, m+1) is the number of degrees of freedom that an m-dimensional flat geometry (point, line, plane, etc.) has when embedded in an n-dimensional Euclidean space.
X(n+1, m+1) is the number of degrees of freedom that an m-ball has when embedded in an n-dimensional Euclidean space. (End)
T(n, k) is also the average number of steps it takes a person to fall off a board of length n+k, if the person starts a random walk at k. - Ruediger Jehn, May 12 2025

			The array T starts in row n=1 with columns m>=1 as:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
   2   4   6   8  10  12  14  16  18  20  22  24  26  28  30
   3   6   9  12  15  18  21  24  27  30  33  36  39  42  45
   4   8  12  16  20  24  28  32  36  40  44  48  52  56  60
   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75
   6  12  18  24  30  36  42  48  54  60  66  72  78  84  90
   7  14  21  28  35  42  49  56  63  70  77  84  91  98 105
   8  16  24  32  40  48  56  64  72  80  88  96 104 112 120
   9  18  27  36  45  54  63  72  81  90  99 108 117 126 135
  10  20  30  40  50  60  70  80  90 100 110 120 130 140 150
The triangle X(n, m) begins
   n\m  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
   1:   1
   2:   2  2
   3:   3  4  3
   4:   4  6  6  4
   5:   5  8  9  8  5
   6:   6 10 12 12 10  6
   7:   7 12 15 16 15 12  7
   8:   8 14 18 20 20 18 14  8
   9:   9 16 21 24 25 24 21 16  9
  10:  10 18 24 28 30 30 28 24 18 10
  11:  11 20 27 32 35 36 35 32 27 20 11
  12:  12 22 30 36 40 42 42 40 36 30 22 12
  13:  13 24 33 40 45 48 49 48 45 40 33 24 13
  14:  14 26 36 44 50 54 56 56 54 50 44 36 26 14
  15:  15 28 39 48 55 60 63 64 63 60 55 48 39 28 15
  ... Formatted by _Wolfdieter Lang_, Dec 02 2014

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 46.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 5-6.

T. D. Noe, Rows n = 1..100 of triangle, flattened
Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10.
G. W. Leibniz, Dissertatio de arte combinatoria, 1666, Leipzig. (in Latin. This triangle appears on p. 208, page 44 of the PDF file).
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9 no. 2 (1997), p. 319-335.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Main diagonal gives squares A000290. Antidiagonal sums are tetrahedral numbers A000292. See A004247 for another version.
Cf. A003989, A003990, A003056, A049581, A000442, A027424, A002260, A033638, A059895, A059896, A059897, A002620.
Cf. also A051776, A067138, A091257, A325821, A331590, A341520, A350066.

/* As triangle */ [[k*(n-k+1): k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jul 12 2019

seq(seq(i*(n-i),i=1..n-1),n=2..10); # Robert Israel, Dec 14 2015

Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)
f[n_] := Table[SeriesCoefficient[E^(x + y) (1+ x - y +x*y-y^2), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11,0]] (* Stefano Spezia, Jul 12 2019 *)

PARI
```
A003991(n,k) = if(k<1 || n<1,0,k*n)
```

Rectangular array: T(n, m) = n*m, n>=1, m>= 1.
Triangle X(n, m) = T(n-m+1, m) = (n-m+1)*m.
Sum_{i=1..n} Sum_{j=1..n} a(n) = A000537(n) [Sum of first n cubes; or n-th triangular number squared.] Determinant of all n X n contiguous subarrays of A003991 is 0. - Gerald McGarvey, Sep 26 2004
G.f. as rectangular array: x*y/((1 - x)^2*(1 - y)^2).
a(n) = i*j, where i=floor((1+sqrt(8n-7))/2), j=n-i*(i-1)/2. - Hieronymus Fischer, Aug 08 2007
As an infinite lower triangular matrix equals A000012 * A002260; where A000012 = (1; 1,1; 1,1,1; ...) and A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Oct 23 2007
As a linear array, the sequence is a(n) = A002260(n)*A004736(n) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012
G.f. as linear array: (x - 3*x^2 + Sum_{k >= 0} ((k+2-x-(k+1)*x^2)*x^((k^2+3*k+4)/2)))/(1-x)^3. - Robert Israel, Dec 14 2015
E.g.f. as triangle: exp(x+y)*(1 + x - y + x*y - y^2). - Stefano Spezia, Jul 12 2019
a(n) = (1/2)*t + (n - 1/4)*t^2 - (1/4)*t^4 - n^2 + n, where t = floor(sqrt(2*n) + 1/2). - Ridouane Oudra, Nov 21 2020
a(n) = A003989(n) * A003990(n) = A059895(n) * A059896(n) = A059895(n)^2 * A059897(n). - Antti Karttunen, Dec 13 2021
T(n,k) = A002620(n+k) - A002620(n-k). - Michel Marcus, Jan 06 2023
T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x < y < z. - Clark Kimberling, Jan 22 2024
E.g.f. as rectangular array: x*y*exp(x+y). - Stefano Spezia, Jun 27 2025

More terms from Michael Somos

A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 9, 9, 9, 9, 9, 7, 8, 10, 8, 8, 8, 8, 10, 8, 9, 7, 15, 7, 7, 7, 15, 7, 9, 10, 8, 10, 12, 10, 10, 12, 10, 8, 10, 11, 15, 7, 15, 25, 15, 25, 15, 7, 15, 11, 12, 14, 12, 10, 12, 18, 18, 12, 10, 12, 14, 12, 13, 25, 21, 25, 15, 25, 11, 25, 15, 25, 21, 25, 13

Offset: 1

Antti Karttunen, Feb 13 2021

Considered as a binary operation on the positive integers, A(x, y) returns the term of the Doudna-sequence from the position that is the sum of the positions of x and y in the same sequence. (This is based on giving the Doudna-sequence an offset of 0, rather than 1 as used in A005940.) - Peter Munn, Feb 14 2021

			The top left 16x16 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10,  11,  12,  13,  14,  15, 16,
   2,  3,  4,  5,  6,  9, 10,  7,  8, 15,  14,  25,  22,  21,  12, 11,
   3,  4,  5,  6,  9,  8, 15, 10,  7, 12,  21,  18,  33,  20,  25, 14,
   4,  5,  6,  9,  8,  7, 12, 15, 10, 25,  20,  27,  28,  35,  18, 21,
   5,  6,  9,  8,  7, 10, 25, 12, 15, 18,  35,  16,  55,  30,  27, 20,
   6,  9,  8,  7, 10, 15, 18, 25, 12, 27,  30,  11,  42,  45,  16, 35,
   7, 10, 15, 12, 25, 18, 11, 16, 27, 14,  49,  20,  77,  50,  21, 24,
   8,  7, 10, 15, 12, 25, 16, 27, 18, 11,  24,  21,  40,  49,  14, 45,
   9,  8,  7, 10, 15, 12, 27, 18, 25, 16,  45,  14,  63,  24,  11, 30,
  10, 15, 12, 25, 18, 27, 14, 11, 16, 21,  50,  35,  70,  75,  20, 49,
  11, 14, 21, 20, 35, 30, 49, 24, 45, 50,  13,  36, 121,  22,  75, 32,
  12, 25, 18, 27, 16, 11, 20, 21, 14, 35,  36,  45,  60, 125,  30, 75,
  13, 22, 33, 28, 55, 42, 77, 40, 63, 70, 121,  60,  17,  98, 105, 48,
  14, 21, 20, 35, 30, 45, 50, 49, 24, 75,  22, 125,  98,  33,  36, 13,
  15, 12, 25, 18, 27, 16, 21, 14, 11, 20,  75,  30, 105,  36,  35, 50,
  16, 11, 14, 21, 20, 35, 24, 45, 30, 49,  32,  75,  48,  13,  50, 81,

Cf. A341511 (the lower triangular section).
Cf. A003961 (main diagonal), A329603 (skewed diagonal).
Cf. A297165 (row 2 and column 2, when started from its term a(1)).
Cf. also A003056, A268715, A341515, A341520, A348041.

up_to = 105;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341510sq(n,k) = A005940(1+A156552(n)+A156552(k));
A341510list(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] = A341510sq(col,(a-(col-1))))); (v); };
v341510 = A341510list(up_to);
A341510(n) = v341510[n];

A(n, k) = A(k, n) = A005940(1 + A156552(n) + A156552(k)).
A(n, n) = A003961(n).
A(n, 2*n) = A(2*n, n) = A329603(n).
A(n, 2) = A(2, n) = A297165(n).

A088698 Replace 1 with 11 in binary representation of n.

Original entry on oeis.org

0, 3, 6, 15, 12, 27, 30, 63, 24, 51, 54, 111, 60, 123, 126, 255, 48, 99, 102, 207, 108, 219, 222, 447, 120, 243, 246, 495, 252, 507, 510, 1023, 96, 195, 198, 399, 204, 411, 414, 831, 216, 435, 438, 879, 444, 891, 894, 1791, 240, 483, 486, 975, 492, 987, 990

Offset: 0

Ralf Stephan, Oct 07 2003

			n=9: 1001 -> 110011 = 51, so a(9) = 51.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Ordered terms plus one are in A048297.
Same sequence sorted into ascending order: A277335, A290258 (without 0).
Cf. A001196, A084471, A048678, A088697.
Main diagonal of A341520, right edge of A341521.

a(n)=if(n<1,0,if(n%2==0,2*a(n/2),4*a((n-1)/2)+3))

def a(n): return int(bin(n)[2:].replace('1', '11'), 2)
print([a(n) for n in range(55)]) # Michael S. Branicky, Feb 20 2021

a(0)=0, a(2n) = 2a(n), a(2n+1) = 4a(n) + 3.

A297164 Permutation of nonnegative integers: a(n) = A156552(A005940(1+n)-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 5, 6, 17, 16, 23, 64, 65, 10, 9, 32, 19, 128, 129, 512, 67, 256, 47, 24, 4097, 20, 4099, 32768, 79, 1024, 11, 18, 31, 14, 29, 4096, 2049, 66, 515, 514, 263, 65536, 2053, 8388608, 524289, 16384, 87, 16777216, 2097153, 70, 524291, 17179869184, 287, 524288, 1037, 8388610, 289, 134217728, 1071, 520, 97, 38, 15, 12, 257, 8192, 63

Offset: 1

Antti Karttunen, Jan 05 2018

nonn

Note the indexing: although the domain starts from 1, the range includes also zero.

Inverse: A297163.
Cf. A005940, A156552.
Cf. also A297166, A341520, A351960.

A005940(n) = { my(p=2,t=1); n--; until(!n\=2, if(n%2, t*=p, p=nextprime(p+1))); t };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A297164(n) = A156552(A005940(1+n)-1); \\ Antti Karttunen, Feb 27 2022

(define (A297164 n) (A156552 (+ -1 (A005940 (+ 1 n)))))

a(n) = A156552(A005940(1+n)-1).

A351960 Square array A(n,k) = A156552(A005940(1+n) + A005940(1+k)), read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 9, 7, 7, 7, 7, 7, 9, 6, 16, 6, 6, 6, 6, 16, 6, 7, 9, 11, 9, 9, 9, 11, 9, 7, 16, 6, 16, 32, 16, 16, 32, 16, 6, 16, 15, 11, 9, 11, 17, 11, 17, 11, 9, 11, 15, 32, 64, 32, 16, 32, 10, 10, 32, 16, 32, 64, 32, 65, 17, 13, 17, 11, 17, 13, 17, 11, 17, 13, 17, 65

Offset: 0

Antti Karttunen, Feb 26 2022

The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.

			The top left corner of the array:
     |n= 0    1    2     3    4     5    6    7    8     9    10    11    12
-----+--------------------------------------------------------------------------
k= 0 |   1,   2,   3,    4,   5,    8,   9,   6,   7,   16,   15,   32,   65,
   1 |   2,   3,   4,    5,   8,    7,  16,   9,   6,   11,   64,   17,   14,
   2 |   3,   4,   5,    8,   7,    6,  11,  16,   9,   32,   13,   10,   35,
   3 |   4,   5,   8,    7,   6,    9,  32,  11,  16,   17,  128,   15,  512,
   4 |   5,   8,   7,    6,   9,   16,  17,  32,  11,   10,   19,   64,   21,
   5 |   8,   7,   6,    9,  16,   11,  10,  17,  32,   15,   18,   13, 1024,
   6 |   9,  16,  11,   32,  17,   10,  13,  64,  15,  128,   23,   18,  129,
   7 |   6,   9,  16,   11,  32,   17,  64,  15,  10,   13,  256,   19,   34,
   8 |   7,   6,   9,   16,  11,   32,  15,  10,  17,   64,   33,  128,   31,
   9 |  16,  11,  32,   17,  10,   15, 128,  13,  64,   19,   12,   33,   20,
  10 |  15,  64,  13,  128,  19,   18,  23, 256,  33,   12,   21,   14,   39,
  11 |  32,  17,  10,   15,  64,   13,  18,  19, 128,   33,   14,   23, 2048,
  12 |  65,  14,  35,  512,  21, 1024, 129,  34,  31,   20,   39, 2048,   25,
  13 | 128,  19,  18,   33, 256,   23,  14,  65,  12,   35,   34,   21, 8192,
  14 |  35, 512,  21, 1024,  31,   34,  27,  20, 129, 2048,   37,   66,  131,
  15 |  64,  13, 128,   19,  18,   33,  12,  23, 256,   65, 1024,   35, 4096,
  16 |  11,  32,  17,   10,  15,   64,  19, 128,  13,   18,   65,  256,   27,

Index entries for sequences related to binary expansion of n

Cf. A005940, A156552.
Cf. A005408 (main diagonal), A297163 (row/column 0).
Cf. also A341510, A341520, A351961, A351962.

up_to = 104;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A351960sq(n,k) = A156552(A005940(1+n)+A005940(1+k));
A351960list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A351960sq(col,(a-(col))))); (v); };
v351960 = A351960list(up_to);
A351960(n) = v351960[1+n];

A351961 Square array A(n,k) = A156552(gcd(A005940(1+n), A005940(1+k))), read by antidiagonals.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 26 2022

The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.

			The top left corner of the array:
   n=  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17
-----|--------------------------------------------------------------
k= 0 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,
   1 | 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,  0,  1,  0,  1,  0,  1,  0,  1,
   2 | 0, 0, 2, 0, 0, 2, 2, 0, 0, 0,  2,  2,  0,  2,  2,  0,  0,  0,
   3 | 0, 1, 0, 3, 0, 1, 0, 3, 0, 1,  0,  3,  0,  1,  0,  3,  0,  1,
   4 | 0, 0, 0, 0, 4, 0, 0, 0, 0, 4,  4,  0,  4,  0,  0,  0,  0,  0,
   5 | 0, 1, 2, 1, 0, 5, 2, 1, 0, 1,  2,  5,  0,  5,  2,  1,  0,  1,
   6 | 0, 0, 2, 0, 0, 2, 6, 0, 0, 0,  2,  2,  0,  6,  6,  0,  0,  0,
   7 | 0, 1, 0, 3, 0, 1, 0, 7, 0, 1,  0,  3,  0,  1,  0,  7,  0,  1,
   8 | 0, 0, 0, 0, 0, 0, 0, 0, 8, 0,  0,  0,  0,  0,  0,  0,  0,  8,
   9 | 0, 1, 0, 1, 4, 1, 0, 1, 0, 9,  4,  1,  4,  1,  0,  1,  0,  1,
  10 | 0, 0, 2, 0, 4, 2, 2, 0, 0, 4, 10,  2,  4,  2,  2,  0,  0,  0,
  11 | 0, 1, 2, 3, 0, 5, 2, 3, 0, 1,  2, 11,  0,  5,  2,  3,  0,  1,
  12 | 0, 0, 0, 0, 4, 0, 0, 0, 0, 4,  4,  0, 12,  0,  0,  0,  0,  0,
  13 | 0, 1, 2, 1, 0, 5, 6, 1, 0, 1,  2,  5,  0, 13,  6,  1,  0,  1,
  14 | 0, 0, 2, 0, 0, 2, 6, 0, 0, 0,  2,  2,  0,  6, 14,  0,  0,  0,
  15 | 0, 1, 0, 3, 0, 1, 0, 7, 0, 1,  0,  3,  0,  1,  0, 15,  0,  1,
  16 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0, 16,  0,
  17 | 0, 1, 0, 1, 0, 1, 0, 1, 8, 1,  0,  1,  0,  1,  0,  1,  0, 17,

Index entries for sequences related to binary expansion of n

Cf. A003989, A005940, A156552.
Cf. A001477 (main diagonal).
Cf. also A341520, A351960, A351962.

up_to = 104; \\ 10439 = binomial(144+1,2)-1
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A351961sq(n,k) = A156552(gcd(A005940(1+n),A005940(1+k)));
A351961list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A351961sq(col,(a-(col))))); (v); };
v351961 = A351961list(up_to);
A351961(n) = v351961[1+n];

For all x, y >= 0, A(x, y) = A(x, A351960(x,y)) = A(A351960(x,y), y).

A351962 Square array A(n,k) = A156552(lcm(A005940(1+n), A005940(1+k))), read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 5, 5, 3, 4, 3, 2, 3, 4, 5, 9, 11, 11, 9, 5, 6, 5, 10, 3, 10, 5, 6, 7, 13, 5, 19, 19, 5, 13, 7, 8, 7, 6, 11, 4, 11, 6, 7, 8, 9, 17, 23, 27, 21, 21, 27, 23, 17, 9, 10, 9, 18, 7, 22, 5, 22, 7, 18, 9, 10, 11, 21, 21, 35, 39, 13, 13, 39, 35, 21, 21, 11, 12, 11, 10, 19, 20, 23, 6, 23, 20, 19, 10, 11, 12

Offset: 0

Antti Karttunen, Feb 26 2022

The indices run as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), etc. The array is symmetric.

			The top left corner of the array:
      n=0   1   2   3   4   5    6    7    8   9   10   11   12
-----|-----------------------------------------------------------
k= 0 |  0,  1,  2,  3,  4,  5,   6,   7,   8,  9,  10,  11,  12,
   1 |  1,  1,  5,  3,  9,  5,  13,   7,  17,  9,  21,  11,  25,
   2 |  2,  5,  2, 11, 10,  5,   6,  23,  18, 21,  10,  11,  26,
   3 |  3,  3, 11,  3, 19, 11,  27,   7,  35, 19,  43,  11,  51,
   4 |  4,  9, 10, 19,  4, 21,  22,  39,  20,  9,  10,  43,  12,
   5 |  5,  5,  5, 11, 21,  5,  13,  23,  37, 21,  21,  11,  53,
   6 |  6, 13,  6, 27, 22, 13,   6,  55,  38, 45,  22,  27,  54,
   7 |  7,  7, 23,  7, 39, 23,  55,   7,  71, 39,  87,  23, 103,
   8 |  8, 17, 18, 35, 20, 37,  38,  71,   8, 41,  42,  75,  44,
   9 |  9,  9, 21, 19,  9, 21,  45,  39,  41,  9,  21,  43,  25,
  10 | 10, 21, 10, 43, 10, 21,  22,  87,  42, 21,  10,  43,  26,
  11 | 11, 11, 11, 11, 43, 11,  27,  23,  75, 43,  43,  11, 107,
  12 | 12, 25, 26, 51, 12, 53,  54, 103,  44, 25,  26, 107,  12,
  13 | 13, 13, 13, 27, 45, 13,  13,  55,  77, 45,  45,  27, 109,
  14 | 14, 29, 14, 59, 46, 29,  14, 119,  78, 93,  46,  59, 110,
  15 | 15, 15, 47, 15, 79, 47, 111,  15, 143, 79, 175,  47, 207,
  16 | 16, 33, 34, 67, 36, 69,  70, 135,  40, 73,  74, 139,  76,

Cf. A003990, A005940, A156552.
Cf. A001477 (main diagonal).
Cf. also A341520, A351960, A351961.

up_to = 104;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A351962sq(n,k) = A156552(lcm(A005940(1+n),A005940(1+k)));
A351962list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A351962sq(col,(a-(col))))); (v); };
v351962 = A351962list(up_to);
A351962(n) = v351962[1+n];

A341521 Triangular array T(n,k) = A156552(A005940(1+n)*A005940(1+k)), read by rows, with n >= 0, 0 <= k <= n.

Original entry on oeis.org

0, 1, 3, 2, 5, 6, 3, 7, 11, 15, 4, 9, 10, 19, 12, 5, 11, 13, 23, 21, 27, 6, 13, 14, 27, 22, 29, 30, 7, 15, 23, 31, 39, 47, 55, 63, 8, 17, 18, 35, 20, 37, 38, 71, 24, 9, 19, 21, 39, 25, 43, 45, 79, 41, 51, 10, 21, 22, 43, 26, 45, 46, 87, 42, 53, 54, 11, 23, 27, 47, 43, 55, 59, 95, 75, 87, 91, 111, 12, 25, 26, 51, 28, 53, 54, 103, 44, 57, 58, 107, 60

Offset: 0

Antti Karttunen, Feb 15 2021

A341520 is the main entry for this dyadic function. See comments there.

			The triangle begins as:
  0,
  1,  3,
  2,  5,  6,
  3,  7, 11, 15,
  4,  9, 10, 19, 12,
  5, 11, 13, 23, 21, 27,
  6, 13, 14, 27, 22, 29, 30,
  7, 15, 23, 31, 39, 47, 55, 63,
  8, 17, 18, 35, 20, 37, 38, 71, 24,
  9, 19, 21, 39, 25, 43, 45, 79, 41, 51,
etc.

The lower triangular region of A341520 read by rows.
Cf. A005940, A156552.
Cf. A001477 (the left edge), A088698 (the right edge).

up_to = 104;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341520sq(n,k) = A156552(A005940(1+n)*A005940(1+k));
A341521list(up_to) = { my(v = vector(1+up_to), i=0); for(n=0,oo, for(k=0,n, i++; if(i > #v, return(v)); v[i] = A341520sq(n,k))); (v); };
v341521 = A341521list(up_to);
A341521(n) = v341521[1+n]; \\ Antti Karttunen, Feb 15 2021

T(n,k) = A341520(n,k).

A341522 a(n) = A156552(3*A005940(1+n)).

Original entry on oeis.org

2, 5, 6, 11, 10, 13, 14, 23, 18, 21, 22, 27, 26, 29, 30, 47, 34, 37, 38, 43, 42, 45, 46, 55, 50, 53, 54, 59, 58, 61, 62, 95, 66, 69, 70, 75, 74, 77, 78, 87, 82, 85, 86, 91, 90, 93, 94, 111, 98, 101, 102, 107, 106, 109, 110, 119, 114, 117, 118, 123, 122, 125, 126, 191, 130, 133, 134, 139, 138, 141, 142, 151, 146, 149

Offset: 0

Antti Karttunen, Feb 15 2021

nonn

Because the least significant 0-bit in A156552-code of any nonzero multiple of 3 is always alone (has 1-bit immediately to its left), it follows that A255068 (= A091067(n+1) - 1) gives these same terms in the ascending order.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for sequences related to binary expansion of n

Row/column 2 of A341520. Permutation of A255068.

Cf. A005940, A007814, A156552, A086799, A014707 (characteristic function).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341522(n) = A156552(3*A005940(1+n));

a(n) = A156552(3*A005940(1+n)).
From Antti Karttunen, Feb 23 2021: (Start)
a(n) = 1 + n + A086799(1+n). - [Conjectured by LODA-miner, and easily seen to be correct]
a(n) = 1+ 2*n + 2^A007814(1+n). - [As the above can be rewritten to this]
(End)

cp's OEIS Frontend

A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

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A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

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A088698 Replace 1 with 11 in binary representation of n.

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A297164 Permutation of nonnegative integers: a(n) = A156552(A005940(1+n)-1).

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A351960 Square array A(n,k) = A156552(A005940(1+n) + A005940(1+k)), read by antidiagonals.

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A351961 Square array A(n,k) = A156552(gcd(A005940(1+n), A005940(1+k))), read by antidiagonals.

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A351962 Square array A(n,k) = A156552(lcm(A005940(1+n), A005940(1+k))), read by antidiagonals.

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