A087061 -id:A087061 - cp's OEIS frontend

A087083 Take unbounded lunar divisors of n as defined in A087029, add them using normal addition. See A087416 for their lunar sum.

45, 44, 42, 39, 35, 30, 24, 17, 9, 495, 4500, 168, 154, 138, 120, 100, 78, 54, 28, 484, 492, 3916, 224, 198, 170, 140, 108, 74, 38, 462, 469, 476, 3276, 258, 220, 180, 138, 94, 48, 429, 435, 441, 447, 2613, 270, 220, 168, 114, 58, 385, 390, 395

Offset: 1

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

D. Applegate, C program for lunar arithmetic and number theory [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 [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

More terms from David Applegate, Nov 07 2003

A087638 Number of lunar primes with <= n digits.

Original entry on oeis.org

0, 18, 99, 1638, 22095, 264312, 3159111, 36694950, 418286661

Offset: 1

Marc LeBrun and N. J. A. Sloane, Oct 26 2003

Partial sums of A087636. - M. F. Hasler, Nov 15 2018

D. Applegate, C program for lunar arithmetic and number theory [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, 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]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087062 (lunar product), A087097 (lunar primes), A087636 (#{n-digit primes}).

A087638(n)=sum(k=2,n,A087636(k)) \\ M. F. Hasler, Nov 15 2018

a(6)-a(9) from David Applegate, Nov 07 2003

A087984 9-ish numbers (A011539) which are not lunar primes (A087097).

Original entry on oeis.org

9, 119, 129, 139, 149, 159, 169, 179, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 229, 239, 249, 259, 269, 279, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 339, 349, 359, 369, 379, 389, 390, 391, 392, 393, 394, 395, 396

Offset: 1

David Applegate and N. J. A. Sloane, Oct 30 2003

Three and four digit 9ish numbers are lunar primes iff the smallest digit is strictly smaller than the first and the last digit. This is no longer true from 10109 = 109 x 109 on (where x = lunar product).

M. F. Hasler, Table of n, a(n) for n = 1..10000
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, preprint, arxiv:1107.1130, 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.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A011539, A087097. A133626 and A134211 are subsequences.

A087984=setminus(A011539, A087097)
A087984=[9]; for(L=3,4,forvec(d=vector(L,i,[i==1,9]),vecmax(d)==9&&vecmin(d)>=min(d[1],d[L])&&A087984=concat(A087984,fromdigits(d)))) \\ terms with < 5 digits

is_A087984(n)=vecmax(digits(n))=9&&!is_A087097(n) \\ (End)

A011539 \ A087097. - M. F. Hasler, Nov 19 2018

A343040 Array T(n, k), n, k >= 0, read by antidiagonals; lunar addition table for the factorial base.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 3, 3, 3, 4, 3, 2, 3, 4, 5, 5, 3, 3, 5, 5, 6, 5, 4, 3, 4, 5, 6, 7, 7, 5, 5, 5, 5, 7, 7, 8, 7, 8, 5, 4, 5, 8, 7, 8, 9, 9, 9, 9, 5, 5, 9, 9, 9, 9, 10, 9, 8, 9, 10, 5, 10, 9, 8, 9, 10, 11, 11, 9, 9, 11, 11, 11, 11, 9, 9, 11, 11, 12, 11, 10, 9, 10, 11, 6, 11, 10, 9, 10, 11, 12

Offset: 0

Rémy Sigrist, Apr 03 2021

The i-th digit of T(n, k) in factorial base is the largest of the i-th digits of n and of k in factorial base.

For n = 0..23, the factorial and primorial base representations of n are the same; hence the date sections for this sequence and for A343044 are the same.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12
  ---+----------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12
    1|   1   1   3   3   5   5   7   7   9   9  11  11  13
    2|   2   3   2   3   4   5   8   9   8   9  10  11  14
    3|   3   3   3   3   5   5   9   9   9   9  11  11  15
    4|   4   5   4   5   4   5  10  11  10  11  10  11  16
    5|   5   5   5   5   5   5  11  11  11  11  11  11  17
    6|   6   7   8   9  10  11   6   7   8   9  10  11  12
    7|   7   7   9   9  11  11   7   7   9   9  11  11  13
    8|   8   9   8   9  10  11   8   9   8   9  10  11  14
    9|   9   9   9   9  11  11   9   9   9   9  11  11  15
   10|  10  11  10  11  10  11  10  11  10  11  10  11  16
   11|  11  11  11  11  11  11  11  11  11  11  11  11  17
   12|  12  13  14  15  16  17  12  13  14  15  16  17  12
Array T(n, k) begins in factorial base:
  n\k|    0    1   10   11   20   21  100  101  110  111  120  121  200
  ---+-----------------------------------------------------------------
    0|    0    1   10   11   20   21  100  101  110  111  120  121  200
    1|    1    1   11   11   21   21  101  101  111  111  121  121  201
   10|   10   11   10   11   20   21  110  111  110  111  120  121  210
   11|   11   11   11   11   21   21  111  111  111  111  121  121  211
   20|   20   21   20   21   20   21  120  121  120  121  120  121  220
   21|   21   21   21   21   21   21  121  121  121  121  121  121  221
  100|  100  101  110  111  120  121  100  101  110  111  120  121  200
  101|  101  101  111  111  121  121  101  101  111  111  121  121  201
  110|  110  111  110  111  120  121  110  111  110  111  120  121  210
  111|  111  111  111  111  121  121  111  111  111  111  121  121  211
  120|  120  121  120  121  120  121  120  121  120  121  120  121  220
  121|  121  121  121  121  121  121  121  121  121  121  121  121  221
  200|  200  201  210  211  220  221  200  201  210  211  220  221  200

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the array for n, k < 6! (where the color is function of T(n, k))
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to factorial base representation

Cf. A087061, A108731, A343040, A343044.

T(n,k) = { my (v=0, f=1); for (r=2, oo, if (n==0 && k==0, return (v), v+=max(n%r, k%r)*f; f*=r; n\=r; k\=r)) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 0) = n.
T(n, n) = n.

A343044 Array T(n, k), n, k >= 0, read by antidiagonals; lunar addition table for the primorial base.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 3, 3, 3, 4, 3, 2, 3, 4, 5, 5, 3, 3, 5, 5, 6, 5, 4, 3, 4, 5, 6, 7, 7, 5, 5, 5, 5, 7, 7, 8, 7, 8, 5, 4, 5, 8, 7, 8, 9, 9, 9, 9, 5, 5, 9, 9, 9, 9, 10, 9, 8, 9, 10, 5, 10, 9, 8, 9, 10, 11, 11, 9, 9, 11, 11, 11, 11, 9, 9, 11, 11, 12, 11, 10, 9, 10, 11, 6, 11, 10, 9, 10, 11, 12

Offset: 0

Rémy Sigrist, Apr 05 2021

The i-th digit of T(n, k) in primorial base is the largest of the i-th digits of n and of k in primorial base.

For n = 0..23, the factorial and primorial base representations of n are the same; hence the date sections for this sequence and for A343040 are the same.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12
  ---+----------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12
    1|   1   1   3   3   5   5   7   7   9   9  11  11  13
    2|   2   3   2   3   4   5   8   9   8   9  10  11  14
    3|   3   3   3   3   5   5   9   9   9   9  11  11  15
    4|   4   5   4   5   4   5  10  11  10  11  10  11  16
    5|   5   5   5   5   5   5  11  11  11  11  11  11  17
    6|   6   7   8   9  10  11   6   7   8   9  10  11  12
    7|   7   7   9   9  11  11   7   7   9   9  11  11  13
    8|   8   9   8   9  10  11   8   9   8   9  10  11  14
    9|   9   9   9   9  11  11   9   9   9   9  11  11  15
   10|  10  11  10  11  10  11  10  11  10  11  10  11  16
   11|  11  11  11  11  11  11  11  11  11  11  11  11  17
   12|  12  13  14  15  16  17  12  13  14  15  16  17  12

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the array for n, k < 2*3*5*7*11 (where the color is function of T(n, k))
Rémy Sigrist, PARI program for A343044
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for sequences related to primorial base

Cf. A087061, A235168, A343040, A343045.

PARI
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See Links section.
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A087416 Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.

Original entry on oeis.org

9, 9, 9, 9, 9, 9, 9, 9, 9, 99, 99, 19, 19, 19, 19, 19, 19, 19, 19, 99, 99, 99, 29, 29, 29, 29, 29, 29, 29, 99, 99, 99, 99, 39, 39, 39, 39, 39, 39, 99, 99, 99, 99, 99, 49, 49, 49, 49, 49, 99, 99, 99, 99, 99, 99, 59, 59, 59, 59, 99, 99, 99, 99, 99, 99, 99, 69, 69, 69

Offset: 1

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

Two comments from David Applegate on lunar perfect numbers, Nov 08 2003: (Start)
If we define a perfect number by "n is lunarly perfect if Sum_{d|n} d == 2*n (both sum and * lunar)", no such numbers exist because 9|n, so the lunar sum of divisors ends in 9, but 2*n ends in 2.
If we define a perfect number by "n is lunarly perfect if lunar Sum_{d|n, d != n} d == n", no such numbers exist. For suppose n is perfect. n != 9 (since 9 is 9's only divisor). Then 9|n and 9 != n, so Sum_{d|n, d!=n} d ends in 9 and thus so does n. But 9ish numbers are not divisible by any single digit < 9. Thus n has no divisors of the same length as n, other than n itself. So Sum_{d|n, d!=n} d is one digit shorter than n. (End)

D. Applegate, C program for lunar arithmetic and number theory [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 [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

More terms from David Applegate, Nov 07 2003

A088469 Number of distinct lunar prime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

David Applegate, Nov 11 2003

nonn

a(n) is the number of lunar primes p that are lunar divisors of n. (Multiplicity is not taken into account. Each prime is counted at most once.)

			10 = 9*90 and 90 is prime. 90 is the only prime divisor of 10, so a(10) = 1.

D. Applegate, C program for lunar arithmetic and number theory [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 [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.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A088471, A088472, A087097.

A088471 Lunar product of distinct lunar prime divisors of n.

Original entry on oeis.org

9, 9, 9, 9, 9, 9, 9, 9, 9, 90, 123456789987654321, 19, 19, 19, 19, 19, 19, 19, 19, 90, 91, 2345678998765432, 29, 29, 29, 29, 29, 29, 29, 90, 91, 92, 34567899876543, 39, 39, 39, 39, 39, 39, 90, 91, 92, 93, 456789987654, 49, 49, 49, 49, 49, 90, 91, 92

Offset: 1

David Applegate, Nov 11 2003

nonn

a(n) = Product_{p is a lunar divisor of n} p. (Each prime appears at most once in this product.)

D. Applegate, C program for lunar arithmetic and number theory [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 [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.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A088469, A088470.

A261684 Array T(n,k) = lunar product n*k (n >= 0, k >= 0) read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 2, 3, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 0, 10, 2, 3, 4, 5, 5, 4, 3, 2, 10, 0, 0, 11, 10, 3, 4, 5, 6, 5, 4, 3, 10, 11, 0

Offset: 0

N. J. A. Sloane, Sep 06 2015

See A087061 for definition. Note that 0+x = x and 9*x = x for all x.

			Lunar multiplication table begins:
0 0 0 0 0 0 ...
0 1 1 1 1 1 ...
0 1 2 2 2 2 ...
0 1 2 3 3 3 ...
0 1 2 3 4 4 ...
0 1 2 3 4 5 ...
....

N. J. A. Sloane, Table of n, a(n) for n = 0..20099
D. Applegate, C program for lunar arithmetic and number theory [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 [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
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. A087061 (addition).
See A087062 for a version that excludes the zero row and column.
Similar to but different from A003983.

# convert decimal to string:
rec := proc(n) local t0,t1,e,l; if n <= 0 then RETURN([[0],1]); fi; t0 := n mod 10; t1 := (n-t0)/10; e := [t0]; l := 1; while t1 <> 0 do t0 := t1 mod 10; t1 := (t1-t0)/10; l := l+1; e := [op(e),t0]; od; RETURN([e,l]); end;
# convert string to decimal:
cer := proc(ep) local i,e,l,t1; e := ep[1]; l := ep[2]; t1 := 0; if l <= 0 then RETURN(t1); fi; for i from 1 to l do t1 := t1+10^(i-1)*e[i]; od; RETURN(t1); end;
# lunar addition:
dadd := proc(m,n) local i,r1,r2,e1,e2,l1,l2,l,l3,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := max(l1,l2); l3 := min(l1,l2); t0 := array(1..l); for i from 1 to l3 do t0[i] := max(e1[i],e2[i]); od; if l>l3 then for i from l3+1 to l do if l1>l2 then t0[i] := e1[i]; else t0[i] := e2[i]; fi; od; fi; cer([t0,l]); end;
# lunar multiplication:
dmul := proc(m,n) local k,i,j,r1,r2,e1,e2,l1,l2,l,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := l1+l2-1; t0 := array(1..l); for i from 1 to l do t0[i] := 0; od; for i from 1 to l2 do for j from 1 to l1 do k := min(e2[i],e1[j]); t0[i+j-1] := max(t0[i+j-1],k); od; od; cer([t0,l]); end;
# to produce the b-file:
M:=199; c:=0; for n from 0 to M do for k from 0 to n do lprint(c,dmul(n-k,k)); c:=c+1; od: od:

A342765 Array T(n, k), n, k > 0, read by antidiagonals; T(n, k) = max(A006530(n), A006530(k)) * T(n/A006530(n), k/A006530(k)) with T(1, 1) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 6, 5, 6, 7, 6, 5, 4, 5, 6, 7, 8, 7, 6, 10, 10, 6, 7, 8, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 9, 12, 14, 10, 10, 14, 12, 9, 10, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 20, 14, 14, 20, 9, 10, 11, 12

Offset: 1

Rémy Sigrist, Apr 02 2021

To compute T(n, k):
- write the prime factors of n and of k in ascending order with multiplicities on two lines, right aligned,
- take the largest prime number in each column and multiply back,
- for example, for T(12, 14):
     12   ->   2 2 3
     14   ->     2 7
               -----
               2 2 7   ->   28 = T(12, 14)
This sequence is closely related to lunar addition (A087061):
- let n and k be two p-smooth numbers,
- let f be the function that associates to a p-smooth number, say m, the unique number whose (p+1)-base digits are prime, nondecreasing and whose product is m,
- let g be the inverse of f,
- then for any p-smooth numbers n and k, T(n, k) = g(f(n) "+" f(k)) where "+" denotes lunar addition in base p+1,
- see A342767 for the corresponding multiplication.

			Array T(n, k) begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12  13  14
  ---+--------------------------------------------------------
    1|   1   2   3   4   5   6   7   8   9  10  11  12  13  14
    2|   2   2   3   4   5   6   7   8   9  10  11  12  13  14
    3|   3   3   3   6   5   6   7  12   9  10  11  12  13  14
    4|   4   4   6   4  10   6  14   8   9  10  22  12  26  14
    5|   5   5   5  10   5  10   7  20  15  10  11  20  13  14
    6|   6   6   6   6  10   6  14  12   9  10  22  12  26  14
    7|   7   7   7  14   7  14   7  28  21  14  11  28  13  14
    8|   8   8  12   8  20  12  28   8  18  20  44  12  52  28
    9|   9   9   9   9  15   9  21  18   9  15  33  18  39  21
   10|  10  10  10  10  10  10  14  20  15  10  22  20  26  14
   11|  11  11  11  22  11  22  11  44  33  22  11  44  13  22
   12|  12  12  12  12  20  12  28  12  18  20  44  12  52  28
   13|  13  13  13  26  13  26  13  52  39  26  13  52  13  26
   14|  14  14  14  14  14  14  14  28  21  14  22  28  26  14

Cf. A001222, A006530, A087061, A342766, A342767.

gpf(n) = if (n==1, 1, my (p=factor(n)[,1]~); p[#p])
T(n, k) = if (n==1 || k==1, max(n, k), my (p=gpf(n), q=gpf(k)); max(p, q)*T(n/p, k/q))

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, 1) = n.
T(n, n) = n.
A001222(T(n, k)) = max(A001222(n), A001222(k)).
A006530(T(n, k)) = max(A006530(n), A006530(k)).

cp's OEIS Frontend

A087083 Take unbounded lunar divisors of n as defined in A087029, add them using normal addition. See A087416 for their lunar sum.

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A087638 Number of lunar primes with <= n digits.

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PARI

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A087984 9-ish numbers (A011539) which are not lunar primes (A087097).

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PARI

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A343040 Array T(n, k), n, k >= 0, read by antidiagonals; lunar addition table for the factorial base.

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PARI

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A343044 Array T(n, k), n, k >= 0, read by antidiagonals; lunar addition table for the primorial base.

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PARI

A087416 Take unbounded lunar divisors of n as defined in A087029, add them using lunar addition. See A087083 for their conventional sum.

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Extensions

A088469 Number of distinct lunar prime divisors of n.

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A088471 Lunar product of distinct lunar prime divisors of n.

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A261684 Array T(n,k) = lunar product n*k (n >= 0, k >= 0) read by antidiagonals.

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