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A306697 revision #20

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).
12
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
1,5
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
LINKS
Eric Weisstein's World of Mathematics, Distributive
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)
EXAMPLE
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
PROG
(PARI) \\ See Links section.
CROSSREFS
Columns (some differing for term 1) and equivalently rows: A003961(3), A000290(4), A045966(5), A045968(7), A045970(11).
Related binary operations: A067138, A059896, A297845/A003991, A329329/A059897.
Sequence in context: A038792 A196416 A329329 * A297845 A183456 A296313
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Mar 05 2019
STATUS
editing