A067399 -id:A067399 - 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.

A190376 a(n) = sum (in ordinary arithmetic) of A067399(k), for k from 2^n to 2^(n+1)-1.

Original entry on oeis.org

1, 4, 12, 31, 75, 175, 393, 864, 1868, 3978, 8394

Offset: 0

N. J. A. Sloane, May 09 2011

I was hoping this would turn out to be a known sequence, in which case we would learn something about the average values of A067399.

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]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A067399, A188548.

read("transforms");
numbralADD := proc(a,b) option remember; ORnos(a,b) ; end proc:
numbralMUL := proc(a,b) option remember; local p,bshf,s ; p := 0 ; bshf := b ; for s from 0 do if bshf mod 2 <> 0 then p := numbralADD(p, 2^s*a ) ; end if; bshf := floor(bshf/2) ; if bshf = 0 then return p; end if; end do; end proc:
isnumbralDiv := proc(n,d) option remember; for e from 0 do if numbralMUL(e,d) = n then return true; elif numbralMUL(e,d) > 2*n then return false; end if; end do: end proc:
numbralDivisors := proc(n) option remember; local d,i; d := {} ; for i from 1 to n do if isnumbralDiv(n,i) then d := d union {i} ; end if; end do: d ; end proc:
A067399 := proc(n) nops(numbralDivisors(n)) ; end proc:
A190376 := proc(n) add(A067399(k),k=2^n..2^(n+1)-1) ; end proc: # R. J. Mathar, May 30 2011

A079500 Number of compositions of the integer n in which the first part is >= the other parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 24, 43, 77, 140, 256, 472, 874, 1628, 3045, 5719, 10780, 20388, 38674, 73562, 140268, 268066, 513350, 984911, 1892875, 3643570, 7023562, 13557020, 26200182, 50691978, 98182666, 190353370, 369393466, 717457656, 1394632365, 2713061899

Offset: 0

Arnold Knopfmacher, Jan 21 2003

nonn

Essentially the same as A007059: a(n) = A007059(n+1).
In lunar arithmetic in base 2, this is the number of lunar divisors of the number 111...1 (with n 1's). E.g., 1111 has a(4) = 5 divisors (see A048888). - N. J. A. Sloane, Feb 23 2011.
First differences of A186537. - N. J. A. Sloane, Feb 23 2011
Number of balanced ordered rooted trees with n non-root nodes (see A048816 for unordered balanced trees); see example. The compositions are obtained from the level sequences by identifying a length-k run of (non-root) levels [t, t+1, t+2, ..., t+k-1] with a part k. - Joerg Arndt, Jul 20 2014

			From _Joerg Arndt_, Dec 29 2012: (Start)
There are a(7)=24 compositions p(1)+p(2)+...+p(m)=7 such that p(k) <= p(1):
[ 1]  [ 1 1 1 1 1 1 1 ]
[ 2]  [ 2 1 1 1 1 1 ]
[ 3]  [ 2 1 1 1 2 ]
[ 4]  [ 2 1 1 2 1 ]
[ 5]  [ 2 1 2 1 1 ]
[ 6]  [ 2 1 2 2 ]
[ 7]  [ 2 2 1 1 1 ]
[ 8]  [ 2 2 1 2 ]
[ 9]  [ 2 2 2 1 ]
[10]  [ 3 1 1 1 1 ]
[11]  [ 3 1 1 2 ]
[12]  [ 3 1 2 1 ]
[13]  [ 3 1 3 ]
[14]  [ 3 2 1 1 ]
[15]  [ 3 2 2 ]
[16]  [ 3 3 1 ]
[17]  [ 4 1 1 1 ]
[18]  [ 4 1 2 ]
[19]  [ 4 2 1 ]
[20]  [ 4 3 ]
[21]  [ 5 1 1 ]
[22]  [ 5 2 ]
[23]  [ 6 1 ]
[24]  [ 7 ]
(End)
From _Joerg Arndt_, Jul 20 2014: (Start)
The a(7) = 24 balanced ordered rooted trees with 7 non-root nodes are, as level sequences (of the pre-order walk):
01:  [ 0 1 1 1 1 1 1 1 ]
02:  [ 0 1 2 1 2 1 2 2 ]
03:  [ 0 1 2 1 2 2 1 2 ]
04:  [ 0 1 2 1 2 2 2 2 ]
05:  [ 0 1 2 2 1 2 1 2 ]
06:  [ 0 1 2 2 1 2 2 2 ]
07:  [ 0 1 2 2 2 1 2 2 ]
08:  [ 0 1 2 2 2 2 1 2 ]
09:  [ 0 1 2 2 2 2 2 2 ]
10:  [ 0 1 2 3 1 2 3 3 ]
11:  [ 0 1 2 3 2 3 2 3 ]
12:  [ 0 1 2 3 2 3 3 3 ]
13:  [ 0 1 2 3 3 1 2 3 ]
14:  [ 0 1 2 3 3 2 3 3 ]
15:  [ 0 1 2 3 3 3 2 3 ]
16:  [ 0 1 2 3 3 3 3 3 ]
17:  [ 0 1 2 3 4 2 3 4 ]
18:  [ 0 1 2 3 4 3 4 4 ]
19:  [ 0 1 2 3 4 4 3 4 ]
20:  [ 0 1 2 3 4 4 4 4 ]
21:  [ 0 1 2 3 4 5 4 5 ]
22:  [ 0 1 2 3 4 5 5 5 ]
23:  [ 0 1 2 3 4 5 6 6 ]
24:  [ 0 1 2 3 4 5 6 7 ]
(End)
From _Gus Wiseman_, Oct 07 2018: (Start)
The a(0) = 1 through a(6) = 14 balanced rooted plane trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((oo)(o))    ((oo)(oo))
                                     ((((oo))))   ((ooo)(o))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o)(oo)))
                                                  (((oo)(o)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))
(End)

Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295.

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 400 terms from T. D. Noe)
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.
Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006).
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.1.
R. Kemp, Balanced ordered trees, Random Structures Algorithms, 5 (1994), pp. 99-121.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A188541.
Cf. A000081 A000311, A001678, A120803, A320154, A320160, A316624, A320169.
Cf. A092921.

M:=101:
t1:=add( (1-x)*x^k/(1-2*x+x^k), k=1..M):
series(t1,x,M-1);
seriestolist(%);
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0, 1,
      `if`(m=0, add(b(n-j, j), j=1..n),
      add(b(n-j, min(n-j, m)), j=1..min(n, m))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, May 01 2014

nn=36;CoefficientList[Series[Sum[x^i/(1-(x-x^(i+1))/(1-x)),{i,0,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 12 2013 *)
b[n_, m_] := b[n, m] = If[n==0, 1, If[m==0, Sum[b[n-j, j], {j, 1, n}], Sum[ b[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

G.f.: (1-z) * Sum_{k>=0} z^k/(1 - 2*z + z^(k+1)).
a(n) = A048888(n) - 1.
This is a subsequence of A067399: a(n) = A067399(2^n-1).
G.f.: -((1 + x^2 + 1/(x-1))/x)*( 1 + x*(x-1)^3*(1-x+x^3)/( Q(0) - x*(x-1)^3*(1-x+x^3)) ), where Q(k) = (x+1)*(2*x-1)*(1-x)^2 + x^(k+2)*(x+x^2+x^3-2*x^4-1 - x^(k+3) + x^(k+5)) - x*(-1+2*x-x^(k+3))*(1-2*x+x^2+x^(k+4)-x^(k+5))*(-1+4*x-5*x^2+2*x^3 - x^(k+2)- x^(k+5) + 2*x^(k+3) - x^(2*k+5) + x^(2*k+6))/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 14 2013
a(n) = Sum_{j=1..n} F(j, n+1-j), where F(n,k) is the n-th k-generalized Fibonacci number A092921(k,n). - Gregory L. Simay, Aug 21 2022

Offset corrected by N. J. A. Sloane, Feb 23 2011
More terms from N. J. A. Sloane, Feb 24 2011
Further edits (required in order to clarify the definition - is the first part >= the rest. or only > the rest? Answer: the former; for the latter, see A007059) by N. J. A. Sloane, May 08 2011

A087029 Number of lunar divisors of n (unbounded version).

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 90, 16, 14, 12, 10, 8, 6, 4, 2, 16, 16, 72, 14, 12, 10, 8, 6, 4, 2, 14, 14, 14, 56, 12, 10, 8, 6, 4, 2, 12, 12, 12, 12, 42, 10, 8, 6, 4, 2, 10, 10, 10, 10, 10, 30, 8, 6, 4, 2, 8, 8, 8, 8, 8, 8, 20, 6, 4, 2, 6, 6, 6, 6, 6, 6, 6, 12, 4, 2, 4, 4, 4, 4

Offset: 1

Marc LeBrun and N. J. A. Sloane, Oct 19 2003

Number of d, 1 <= d < infinity, such that there exists an e, 1 <= e < infinity, with d*e = n, where * is lunar multiplication.

			The 18 divisors of 10 are 1, 2, ..., 9, 10, 20, 30, ..., 90, so a(10) = 18.

D. Applegate, Table of n, a(n) for n = 1..100000
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arxiv:1107.1130 [math-NT], July 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.
Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087062 (lunar product).
Cf. A087028, A087083, A186443, A186510. See A189506 for the actual divisors.
See A067399 for the base-2 version.

(Uses programs from A087062. This crude program is valid for n <= 99.) dd2 := proc(n) local t1,t2,i,j; t1 := []; for i from 1 to 99 do for j from i to 99 do if dmul(i,j) = n then t1 := [op(t1),i,j]; fi; od; od; t1 := convert(t1,set); t2 := sort(convert(t1,list)); nops(t2); end;

A087029(n)=#A189506_row(n) \\ To be optimized. - M. F. Hasler, Nov 15 2018

More terms from David Applegate, Nov 07 2003

Minor edits by M. F. Hasler, Nov 15 2018

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

A188548 The sum of the divisors of n in base-2 lunar arithmetic.

Original entry on oeis.org

1, 11, 11, 111, 101, 111, 111, 1111, 1001, 1111, 1011, 1111, 1101, 1111, 1111, 11111, 10001, 11011, 10011, 11111, 10101, 11111, 10111, 11111, 11001, 11111, 11011, 11111, 11101, 11111, 11111, 111111, 100001, 110011, 100011, 111111, 100101, 110111, 100111, 111111, 101001, 111111, 101011, 111111, 101101, 111111, 101111, 111111, 110001, 111011, 110011, 111111

Offset: 1

N. J. A. Sloane, Apr 04 2011

More precisely, in base-2 lunar arithmetic, the lunar sum of the lunar divisors of the n-th nonzero binary number.

Theorem: a(n) = binary representation of n iff n is odd.

			The 4th binary number is 100 which has lunar divisors 1, 10, 100, whose lunar sum is 111, so a(4)=111.
The 5th binary number is 101 which has lunar divisors 1 and 101, whose lunar sum is 101, so a(5)=101.
It might be tempting to conjecture that if n is even then a(n) = 111...111, but a(18)=11011 shows that this is false (see A190149).

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]
N. J. A. Sloane, Table giving n (written in base 10), n (written in base 2), a(n) (written in base 2), a(n) (written in base 10)
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A067399 (number of divisors), A190149, A190632.

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.

A190149 Even numbers n (written in binary) such that in base-2 lunar arithmetic, the sum of the divisors of n is a number containing a 0 (in binary).

Original entry on oeis.org

10010, 100010, 100110, 110010, 1000010, 1000100, 1000110, 1001010, 1001110, 1010010, 1100010, 1100110, 1110010, 10000010, 10000100, 10000110, 10001010, 10001100, 10001110, 10010010, 10010110, 10011010, 10011110, 10100010, 10100110, 10110010, 11000010, 11000100, 11000110, 11001010, 11001110, 11010010, 11100010, 11100110, 11110010, 100000010, 100000100

Offset: 1

N. J. A. Sloane, May 05 2011

As remarked in A188548, if n is even then most of the time A188548(n) = 111...111 (that is, a number of the form 2^k-1). This sequence lists the exceptions.

			In base-2 lunar arithmetic, the divisors of 10010 are 1, 10, 1001 and 10010, whose sum is 11011.

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, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A188548, A067399. See A190150 and A190151 for the base-10 representation of these numbers.

A187141 Number of lunar divisors of n in base 3.

Original entry on oeis.org

2, 1, 4, 6, 2, 2, 2, 2, 6, 6, 2, 12, 14, 3, 4, 4, 4, 3, 2, 2, 4, 3, 2, 4, 4, 3, 8, 6, 2, 12, 10, 2, 4, 2, 2, 18, 10, 2, 28, 34, 5, 6, 7, 7, 6, 2, 2, 8, 7, 5, 8, 8, 6, 4, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 6, 5, 2, 4, 5, 2, 6, 2, 2, 8, 7, 2, 6, 6, 6, 10, 6, 2, 12, 10, 2, 4, 2, 2, 18, 14, 3, 20, 18, 2, 4, 2, 2, 6, 4, 4, 4, 2, 2, 4, 2, 2, 24, 10, 2, 20, 26, 4, 4, 4, 4, 42, 18, 2, 68

Offset: 1

N. J. A. Sloane, Mar 05 2011

			If written as a triangle, with rows of length 2*3^i:
2, 1,
4, 6, 2, 2, 2, 2,
6, 6, 2, 12, 14, 3, 4, 4, 4, 3, 2, 2, 4, 3, 2, 4, 4, 3,
8, 6, 2, 12, 10, 2, 4, 2, 2, 18, 10, 2, 28, 34, 5, 6, 7, 7, 6, 2, 2, 8, 7, 5, 8, 8, 6, 4, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 6, 5, 2, 4, 5, 2, 6, 2, 2, 8, 7, 2, 6, 6, 6,
10, 6, 2, 12, 10, 2, 4, 2, 2, 18, 14, 3, 20, 18, 2, 4, 2, 2, 6, 4, 4, 4, 2, 2, 4, 2, 2, 24, 10, 2, 20, 26, 4, 4, 4, 4, 42, 18, 2, 6, ...

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]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A007089. This is the base-3 analog of A067399.

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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A190376 a(n) = sum (in ordinary arithmetic) of A067399(k), for k from 2^n to 2^(n+1)-1.

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A079500 Number of compositions of the integer n in which the first part is >= the other parts.

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A087029 Number of lunar divisors of n (unbounded version).

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A067398 Squares in OR-numbral arithmetic.

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A188548 The sum of the divisors of n in base-2 lunar 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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A190149 Even numbers n (written in binary) such that in base-2 lunar arithmetic, the sum of the divisors of n is a number containing a 0 (in binary).