A067138 -id:A067138 - cp's OEIS frontend

A348135 Irregular triangle T(n, k), n > 0, k = 1..A067399(n), read by rows; the n-th row gives, in ascending order, the distinct integers k such that A067138(k, m) = n for some m.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 3, 7, 1, 2, 4, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 3, 6, 7, 14, 1, 3, 5, 7, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 5, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24

Offset: 1

Rémy Sigrist, Oct 02 2021

The n-th row corresponds to the divisors of n in OR-numbral arithmetic.

			The triangle starts:
      1:   [1]
      2:   [1, 2]
      3:   [1, 3]
      4:   [1, 2, 4]
      5:   [1, 5]
      6:   [1, 2, 3, 6]
      7:   [1, 3, 7]
      8:   [1, 2, 4, 8]
      9:   [1, 9]
     10:   [1, 2, 5, 10]
     11:   [1, 11]
     12:   [1, 2, 3, 4, 6, 12]
     13:   [1, 13]
     14:   [1, 2, 3, 6, 7, 14]
     15:   [1, 3, 5, 7, 15]
     16:   [1, 2, 4, 8, 16]

Rémy Sigrist, Table of n, a(n) for n = 1..7412 (first 1024 rows flattened)
Rémy Sigrist, PARI program for A348135
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A067138, A067399, A346795.

PARI
```
See Links section.
```

T(n, 1) = 1.

T(n, A067399(n)) = n.

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

Original entry on oeis.org

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

A067139 Irreducible elements in OR-numbral arithmetic.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 13, 17, 19, 23, 25, 29, 33, 35, 37, 39, 41, 43, 49, 53, 57, 65, 67, 69, 71, 75, 77, 79, 81, 83, 87, 89, 93, 97, 101, 105, 107, 113, 117, 121, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 157, 159, 161, 163, 167, 169, 171, 177, 179

Offset: 1

Jens Voß, Jan 02 2002

Numbers m such that there is no number d in the range 1 < d < m with d*k = m for any 1 < k < m, where * is defined in A066376.
See A048888 for the definition of OR-numbral arithmetic. Note that 2 is the only prime element in OR-numbral arithmetic; for all other nonunit irreducibles x there exist numbers a and b not divisible by x such that x is a divisor of a * b.
Numbers m such that A066376(m) = 1.
1 together with primes in lunar arithmetic base 2. - N. J. A. Sloane, Aug 14 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..500
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A003986, A067138, A048888, A007059.

See A169912 for the number of elements that are n bits long - N. J. A. Sloane, Aug 31 2010. See A171000 for the binary expansions.

import Data.List (elemIndices)
a067139 n = a067139_list !! (n-1)
a067139_list = 1 : map (+ 1) (elemIndices 1 a066376_list)
-- Reinhard Zumkeller, Mar 01 2013

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe and Joshua Zucker, Jun 12 2007

A306697 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is obtained by applying a Minkowski sum to sets related to the Fermi-Dirac factorizations of n and of k (see Comments for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 30, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81

Offset: 1

Rémy Sigrist, Mar 05 2019

For any m > 0:
- let F(m) be the set of distinct Fermi-Dirac primes (A050376) with product m,
- for any i >=0 0 and j >= 0, let f(prime(i+1)^(2^i)) be the lattice point with coordinates X=i and Y=j (where prime(k) denotes the k-th prime number),
- f establishes a bijection from the Fermi-Dirac primes to the lattice points with nonnegative coordinates,
- let P(m) = { f(p) | p in F(m) },
- P establishes a bijection from the nonnegative integers to the set, say L, of finite sets of lattice points with nonnegative coordinates,
- let Q be the inverse of P,
- for any n > 0 and k > 0:
    T(n, k) = Q(P(n) + P(k))
              where "+" denotes the Minkowski addition on L.
This sequence has similarities with A297845, and their data sections almost match; T(6, 6) = 30, however A297845(6, 6) = 90.
This sequence has similarities with A067138; here we work on dimension 2, there in dimension 1.
This sequence as a binary operation distributes over A059896, whereas A297845 distributes over multiplication (A003991) and A329329 distributes over A059897. See the comment in A329329 for further description of the relationship between these sequences. - Peter Munn, Dec 19 2019

			Array T(n, k) begins:
  n\k|  1   2   3    4    5    6    7     8     9    10    11    12
  ---+-------------------------------------------------------------
    1|  1   1   1    1    1    1    1     1     1     1     1     1
    2|  1   2   3    4    5    6    7     8     9    10    11    12
    3|  1   3   5    9    7   15   11    27    25    21    13    45
    4|  1   4   9   16   25   36   49    64    81   100   121   144
    5|  1   5   7   25   11   35   13   125    49    55    17   175
    6|  1   6  15   36   35   30   77   216   225   210   143   540
    7|  1   7  11   49   13   77   17   343   121    91    19   539
    8|  1   8  27   64  125  216  343   128   729  1000  1331  1728
    9|  1   9  25   81   49  225  121   729   625   441   169  2025
   10|  1  10  21  100   55  210   91  1000   441   110   187  2100
   11|  1  11  13  121   17  143   19  1331   169   187    23  1573
   12|  1  12  45  144  175  540  539  1728  2025  2100  1573   720

Rémy Sigrist, PARI program for A306697
OEIS Wiki, "Fermi-Dirac representation" of n
Eric Weisstein's World of Mathematics, Distributive
Wikipedia, Minkowski addition

Columns (some differing for term 1) and equivalently rows: A003961(3), A000290(4), A045966(5), A045968(7), A045970(11).
Cf. A000040, A001221, A019565, A064547, A225546, A329050.
Related binary operations: A067138, A059896, A297845/A003991, A329329/A059897.

PARI
```
\\ See Links section.
```

For any m > 0, n > 0, k > 0, i >= 0, j >= 0:
- T(n, k) = T(k, n) (T is commutative),
- T(m, T(n, k)) = T(T(m, n), k) (T is associative),
- T(n, 1) = 1 (1 is an absorbing element for T),
- T(n, 2) = n (2 is an identity element for T),
- T(n, 3) = A003961(n),
- T(n, 4) = n^2 (A000290),
- T(n, 5) = A357852(n),
- T(n, 7) = A045968(n) (when n > 1),
- T(n, 11) = A045970(n) (when n > 1),
- T(n, 2^(2^i)) = n^(2^i),
- T(2^i, 2^j) = 2^A067138(i, j),
- T(A019565(i), A019565(j)) = A019565(A067138(i, j)),
- T(A000040(n), A000040(k)) = A000040(n + k - 1),
- T(2^(2^i), 2^(2^j)) = 2^(2^(i + j)),
- A001221(T(n, k)) <= A001221(n) * A001221(k),
- A064547(T(n, k)) <= A064547(n) * A064547(k).
From Peter Munn, Dec 05 2019:(Start)
T(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).
Equivalently, T(prime(i_1 - 1)^(2^(j_1)), prime(i_2 - 1)^(2^(j_2))) = prime(i_1+i_2 - 1)^(2^(j_1+j_2)), where prime(i) = A000040(i).
T(A059896(i,j), k) = A059896(T(i,k), T(j,k)) (T distributes over A059896).
T(A019565(i), 2^j) = A019565(i)^j.
T(A225546(i), A225546(j)) = A225546(T(i,j)).
(End)

A067399 Number of divisors of n in OR-numbral arithmetic.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 4, 2, 4, 2, 6, 2, 6, 5, 5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 9, 2, 10, 8, 6, 2, 4, 2, 6, 2, 4, 2, 8, 2, 6, 2, 6, 4, 4, 4, 10, 2, 4, 4, 6, 2, 8, 4, 12, 2, 4, 4, 15, 4, 16, 14, 7, 2, 4, 2, 6, 2, 4, 2, 8, 3, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 9, 5, 4, 2, 8, 2, 8, 4, 6, 2, 8, 6, 12, 2, 4, 4, 6

Offset: 1

Jens Voß, Jan 23 2002

nonn

See A048888 for the definition of OR-numbral arithmetic. The example shows that this sequence is not multiplicative.

In other words, number of lunar divisors of n in base 2.

			a(15)=5 since [15] has the 5 OR-numbral divisors [1], [3], [5], [7] and [15].
If written as a triangle with rows of lengths 1,2,4,8,16,...:
1,
2, 2,
3, 2, 4, 3,
4, 2, 4, 2, 6, 2, 6, 5,
5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 9, 2, 10, 8,
6, 2, 4, 2, 6, 2, 4, 2, 8, 2, 6, 2, 6, 4, 4, 4, 10, 2, 4, 4, 6, 2, 8, 4, 12, 2, 4, 4, 15, 4, 16, 14,
...,
the last terms in each row give A079500(n). The penultimate terms in the rows give 2*A079500(n-1). - _N. J. A. Sloane_, Mar 05 2011

N. J. A. Sloane, Table of n, a(n) for n = 1..1024
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Index entries for sequences related to dismal (or lunar) arithmetic

A079500 is the subsequence a(2^k-1). - N. J. A. Sloane, Feb 23 2011
Cf. A003986, A007059, A048888, A067138, A067139, A067398, A067400, A067401.
See A188548 for the sum of the divisors.

A067398 Squares in OR-numbral arithmetic.

Original entry on oeis.org

0, 1, 4, 7, 16, 21, 28, 31, 64, 73, 84, 95, 112, 125, 124, 127, 256, 273, 292, 311, 336, 341, 380, 383, 448, 473, 500, 511, 496, 509, 508, 511, 1024, 1057, 1092, 1127, 1168, 1205, 1244, 1279, 1344, 1385, 1364, 1407, 1520, 1533, 1532, 1535, 1792, 1841, 1892

Offset: 0

Jens Voß, Jan 23 2002

nonn

See A048888 for the definition of OR-numbral arithmetic.
Or, squares in lunar arithmetic base 2, written in base 10. - N. J. A. Sloane, Oct 02 2010
This sequence is not multiplicative; for example a(15) = 127 != 7 * 21 = a(3) * a(5). It is totally OR-numbral multiplicative: a([n] * [m]) = [a(n)] * [a(m)] in OR-numbral arithmetic. - Franklin T. Adams-Watters, Oct 27 2006

			A067398(5) = 21 since [5] * [5] = [21] in OR-numbral arithmetic.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A003986, A007059, A048888, A067138, A067139, A067399, A067400, A067401.

a067398 :: Integer -> Integer
a067398 0 = 0
a067398 n = orm n n where
   orm 1 v = v
   orm u v = orm (shiftR u 1) (shiftL v 1) .|. if odd u then v else 0
-- Reinhard Zumkeller, Mar 01 2013

A067400 Non-uniquely factorizable OR-numbrals, i.e., numbrals for which there exist more than one different factorizations into irreducible factors (modulo order).

Original entry on oeis.org

15, 30, 31, 60, 62, 63, 85, 95, 111, 120, 123, 124, 125, 126, 127, 170, 175, 190, 191, 207, 222, 223, 239, 240, 243, 245, 246, 247, 248, 250, 251, 252, 253, 254, 255, 340, 341, 350, 351, 367, 379, 380, 381, 382, 383, 399, 414, 415, 443, 444, 445, 446, 447

Offset: 1

Jens Voß, Jan 24 2002

nonn

See A048888 for the definition of OR-numbral arithmetic.

			15 is in A067400 since [15] = [3] * [5] = [3]^3 and [3] and [5] are irreducible.

Cf. A003986, A007059, A048888, A067138, A067139, A067398, A067399, A067401.

A067401 Minimal non-uniquely factorizable OR-numbrals, i.e., numbrals that are not uniquely factorizable but for which all proper divisors are.

Original entry on oeis.org

15, 85, 95, 111, 123, 125, 175, 191, 207, 223, 239, 243, 245, 247, 251, 253, 351, 367, 379, 381, 399, 415, 443, 445, 447, 463, 483, 487, 493, 499, 501, 507, 585, 603, 621, 631, 639, 685, 687, 701, 725, 729, 731, 735, 757, 763, 783, 799, 827, 831, 873, 877

Offset: 1

Jens Voß, Jan 24 2002

nonn

See A048888 for the definition of OR-numbral arithmetic.

			15 is in A067401 since [15] = [3] * [5] = [3]^3 all divisors of [15] are uniquely factorizable.

Cf. A003986, A007059, A048888, A067138, A067139, A067398, A067399, A067400.

A342639 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = g(f(n) + f(k)) where g maps the complement, say s, of a finite set of nonnegative integers to the value Sum_{e >= 0 not in s} 2^e, f is the inverse of g, and "+" denotes the Minkowski sum.

Original entry on oeis.org

0, 1, 1, 0, 3, 0, 3, 1, 1, 3, 0, 7, 2, 7, 0, 1, 1, 3, 3, 1, 1, 0, 3, 0, 15, 0, 3, 0, 7, 1, 5, 3, 3, 5, 1, 7, 0, 15, 2, 7, 0, 7, 2, 15, 0, 1, 1, 7, 3, 1, 1, 3, 7, 1, 1, 0, 3, 0, 31, 4, 11, 4, 31, 0, 3, 0, 3, 1, 1, 3, 7, 5, 5, 7, 3, 1, 1, 3, 0, 7, 2, 7, 0, 15, 6, 15, 0, 7, 2, 7, 0

Offset: 0

Rémy Sigrist, Mar 17 2021

In other words:
- we consider the set S of sets s of nonnegative integers whose complement is finite,
- the function g encodes the "missing integers" in binary:
          g(A001477 \ {1, 4}) = 2^1 + 2^4 = 18
- the function f is the inverse of g:
          f(42) = f(2^1 + 2^3 + 2^5) = A001477 \ {1, 3, 5},
- the Minkowski sum of two sets, say U and V, is the set of sums u+v where u belongs to U and v belongs to V,
- the Minkowski sum is stable over S,
- and T provides an encoding for this operation.
This sequence has connections with A067138; here we consider complements of finite sets of nonnegative integers, there finite sets of nonnegative integers.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6    7   8   9  10  11  12  13  14   15
  ---+------------------------------------------------------------------
    0|   0   1   0   3   0   1   0    7   0   1   0   3   0   1   0   15
    1|   1   3   1   7   1   3   1   15   1   3   1   7   1   3   1   31
    2|   0   1   2   3   0   5   2    7   0   1   2  11   0   5   2   15
    3|   3   7   3  15   3   7   3   31   3   7   3  15   3   7   3   63
    4|   0   1   0   3   0   1   4    7   0   1   0   3   0   9   4   15
    5|   1   3   5   7   1  11   5   15   1   3   5  23   1  11   5   31
    6|   0   1   2   3   4   5   6    7   0   9   2  11   4  13   6   15
    7|   7  15   7  31   7  15   7   63   7  15   7  31   7  15   7  127
    8|   0   1   0   3   0   1   0    7   0   1   0   3   0   1   8   15
    9|   1   3   1   7   1   3   9   15   1   3   1   7   1  19   9   31
   10|   0   1   2   3   0   5   2    7   0   1  10  11   0   5  10   15
   11|   3   7  11  15   3  23  11   31   3   7  11  47   3  23  11   63
   12|   0   1   0   3   0   1   4    7   0   1   0   3   8   9  12   15
   13|   1   3   5   7   9  11  13   15   1  19   5  23   9  27  13   31
   14|   0   1   2   3   4   5   6    7   8   9  10  11  12  13  14   15
   15|  15  31  15  63  15  31  15  127  15  31  15  63  15  31  15  255

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Wikipedia, Minkowski addition

Cf. A038712, A067138, A133457, A135481, A342640, A342641, A342642.

T(n,k) = { my (v=0); for (x=0, #binary(n)+#binary(k), my (f=0); for (y=0, x, if (!bittest(n,y) && !bittest(k,x-y), f=1; break)); if (!f, v+=2^x)); return (v) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = A135481(n).
T(n, 1) = A038712(n+1).
T(2^n-1, 2^k-1) = 2^(n+k)-1.
T(n, n) = A342640(n).

A066376 Number of []-divisors d <= n such that there is another []-divisor d' < n with d [*] d' = n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 3, 1, 3, 1, 5, 1, 5, 4, 4, 1, 3, 1, 5, 2, 3, 1, 7, 1, 3, 3, 8, 1, 9, 7, 5, 1, 3, 1, 5, 1, 3, 1, 7, 1, 5, 1, 5, 3, 3, 3, 9, 1, 3, 3, 5, 1, 7, 3, 11, 1, 3, 3, 14, 3, 15, 13, 6, 1, 3, 1, 5, 1, 3, 1, 7, 2, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 8, 4, 3, 1, 7, 1, 7, 3, 5, 1, 7, 5, 11, 1

Offset: 1

Marc LeBrun, Dec 22 2001

Define [+] to be binary bitwise inclusive-OR and let [*] denote the shift-and-[+] product. ([+] is usually simply called OR.) Note that [*] is commutative, associative, and distributes over [+]. If x [*] y = z, we say x and y are [*]-divisors of z.

			14 has 5 [*]-divisors: 1, 2, 3, 6, 7, since for example 2 [*] 7 = 10 [*] 111 = 1110 OR 0000 = 1110; and 3 [*] 6 = 11 [*] 110 = 1100 OR 0110 = 1110.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A067139 ("primes").

See A003986 for a table of [+] sums, A067138 for a table of [*] products.

import Data.Bits (Bits, (.|.), shiftL, shiftR)
a066376 :: Int -> Int
a066376 n = length [d | d <- [1..n-1], any ((== n) . (orm d)) [1..n]] where
   orm 1 v = v
   orm u v = orm (shiftR u 1) (shiftL v 1) .|. if odd u then v else 0
-- Reinhard Zumkeller, Mar 01 2013

Edited by N. J. A. Sloane, Dec 13 2021

Name corrected by Sean A. Irvine, Oct 10 2023

cp's OEIS Frontend

A348135 Irregular triangle T(n, k), n > 0, k = 1..A067399(n), read by rows; the n-th row gives, in ascending order, the distinct integers k such that A067138(k, m) = n for some m.

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

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Magma

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A067139 Irreducible elements in OR-numbral arithmetic.

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A306697 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: T(n, k) is obtained by applying a Minkowski sum to sets related to the Fermi-Dirac factorizations of n and of k (see Comments for precise definition).

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A067399 Number of divisors of n in OR-numbral arithmetic.

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A067398 Squares in OR-numbral arithmetic.

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A067400 Non-uniquely factorizable OR-numbrals, i.e., numbrals for which there exist more than one different factorizations into irreducible factors (modulo order).

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A067401 Minimal non-uniquely factorizable OR-numbrals, i.e., numbrals that are not uniquely factorizable but for which all proper divisors are.

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A066376 Number of []-divisors d <= n such that there is another []-divisor d' < n with d [*] d' = n.