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A306697
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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).
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12
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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
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OFFSET
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1,5
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COMMENTS
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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
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LINKS
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FORMULA
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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, 5) = A045966(n) (when n > 1),
- 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^(2^i), 2^(2^j)) = 2^(2^(i + j)),
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).
(End)
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EXAMPLE
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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
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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