A343033 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime exponents of numbers (see Comments section for precise definition).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 5, 2, 1, 1, 5, 3, 3, 5, 1, 1, 6, 7, 4, 7, 6, 1, 1, 7, 15, 5, 5, 15, 7, 1, 1, 2, 11, 6, 11, 6, 11, 2, 1, 1, 3, 3, 7, 35, 35, 7, 3, 3, 1, 1, 10, 5, 4, 13, 30, 13, 4, 5, 10, 1, 1, 11, 21, 9, 5, 77, 77, 5, 9, 21, 11, 1

Offset: 1

Author

Rémy Sigrist and N. J. A. Sloane, Apr 03 2021

Keywords

• nonn
• tabl

Comments

To compute T(n, k):

- write the prime exponents of n and of k on two lines, right aligned (these lines correspond to rows of A067255 in reversed order),

- to "multiply" two prime numbers: take the smallest,

- to "add" two prime numbers: take the largest,

- for example, for T(12, 14):

(11 7 5 3 2)

12 --> 1 2

14 --> x 1 0 0 1

---------

1 1

0 0

0 0

+ 1 1

-----------

1 1 0 1 1 --> 462 = T(12, 14)

This sequence is closely related to lunar multiplication (A087062):

- for any b > 1, let S_b be the set of nonnegative integers m such that A051903(m)< b,

- there is a natural bijection f from S_b to the set of nonnegative integers:

f(Product_{k >= 0} prime(k)^d(k)) = Sum_{k >= 0} d(k) * b^k,

- let g be the inverse of f,

- then for any numbers n and k in S_b, we have:

T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base b,

- the corresponding addition table is A003990.

Examples

			Array T(n, k) begins:
  n\k|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
  ----  -  --  --  --  --  ---  ---  --  --  ---  ---  ---  ---  ---
    1|  1   1   1   1   1    1    1   1   1    1    1    1    1    1
    2|  1   2   3   2   5    6    7   2   3   10   11    6   13   14  --> A007947
    3|  1   3   5   3   7   15   11   3   5   21   13   15   17   33  --> A328915
    4|  1   2   3   4   5    6    7   4   9   10   11   12   13   14  --> A007948
    5|  1   5   7   5  11   35   13   5   7   55   17   35   19   65
    6|  1   6  15   6  35   30   77   6  15  210  143   30  221  462
    7|  1   7  11   7  13   77   17   7  11   91   19   77   23  119
    8|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
    9|  1   3   5   9   7   15   11   9  25   21   13   45   17   33
   10|  1  10  21  10  55  210   91  10  21  110  187  210  247  910
   11|  1  11  13  11  17  143   19  11  13  187   23  143   29  209
   12|  1   6  15  12  35   30   77  12  45  210  143   60  221  462
   13|  1  13  17  13  19  221   23  13  17  247   29  221   31  299
   14|  1  14  33  14  65  462  119  14  33  910  209  462  299  238
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10011
• Index entries for sequences related to dismal (or lunar) arithmetic

Crossrefs

Cf. A003990, A007947, A007948, A051903, A067255, A087062, A328915, A342767, A343035.

Programs

PARI

T(n,k) = { my (r=1, pp=factor(n)[,1]~, qq=factor(k)[,1]~); for (i=1, #pp, for (j=1, #qq, my (p=prime(primepi(pp[i])+primepi(qq[j])-1), v=valuation(r, p), w=min(valuation(n, pp[i]), valuation(k, qq[j]))); if (w>v, r*=p^(w-v)))); r }

Formula

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

T(n, 1) = 1.

T(n, 2) = A007947(n).

T(n, 3) = A328915(n).

T(n, 4) = A007948(n).

T(n, n) = A343035(n).

A051903(T(n, k)) = min(A051903(n), A051903(k)).