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).

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

Author

Rémy Sigrist, Mar 05 2019

Keywords

• nonn
• tabl

Comments

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

Examples

			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
		

Links

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

Crossrefs

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.

Programs

PARI

\\ See Links section.

Formula

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)