This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306697 #23 Feb 16 2025 08:33:55
%S A306697 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,
%T A306697 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,
%U A306697 64,13,30,13,64,25,10,1,1,11,21,81,125,77,77,125,81
%N 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).
%C A306697 For any m > 0:
%C A306697 - let F(m) be the set of distinct Fermi-Dirac primes (A050376) with product m,
%C A306697 - 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),
%C A306697 - f establishes a bijection from the Fermi-Dirac primes to the lattice points with nonnegative coordinates,
%C A306697 - let P(m) = { f(p) | p in F(m) },
%C A306697 - P establishes a bijection from the nonnegative integers to the set, say L, of finite sets of lattice points with nonnegative coordinates,
%C A306697 - let Q be the inverse of P,
%C A306697 - for any n > 0 and k > 0:
%C A306697 T(n, k) = Q(P(n) + P(k))
%C A306697 where "+" denotes the Minkowski addition on L.
%C A306697 This sequence has similarities with A297845, and their data sections almost match; T(6, 6) = 30, however A297845(6, 6) = 90.
%C A306697 This sequence has similarities with A067138; here we work on dimension 2, there in dimension 1.
%C A306697 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
%H A306697 Rémy Sigrist, <a href="/A306697/a306697.gp.txt">PARI program for A306697</a>
%H A306697 OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>
%H A306697 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Distributive.html">Distributive</a>
%H A306697 Wikipedia, <a href="https://en.wikipedia.org/wiki/Minkowski_addition">Minkowski addition</a>
%F A306697 For any m > 0, n > 0, k > 0, i >= 0, j >= 0:
%F A306697 - T(n, k) = T(k, n) (T is commutative),
%F A306697 - T(m, T(n, k)) = T(T(m, n), k) (T is associative),
%F A306697 - T(n, 1) = 1 (1 is an absorbing element for T),
%F A306697 - T(n, 2) = n (2 is an identity element for T),
%F A306697 - T(n, 3) = A003961(n),
%F A306697 - T(n, 4) = n^2 (A000290),
%F A306697 - T(n, 5) = A357852(n),
%F A306697 - T(n, 7) = A045968(n) (when n > 1),
%F A306697 - T(n, 11) = A045970(n) (when n > 1),
%F A306697 - T(n, 2^(2^i)) = n^(2^i),
%F A306697 - T(2^i, 2^j) = 2^A067138(i, j),
%F A306697 - T(A019565(i), A019565(j)) = A019565(A067138(i, j)),
%F A306697 - T(A000040(n), A000040(k)) = A000040(n + k - 1),
%F A306697 - T(2^(2^i), 2^(2^j)) = 2^(2^(i + j)),
%F A306697 - A001221(T(n, k)) <= A001221(n) * A001221(k),
%F A306697 - A064547(T(n, k)) <= A064547(n) * A064547(k).
%F A306697 From _Peter Munn_, Dec 05 2019:(Start)
%F A306697 T(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).
%F A306697 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).
%F A306697 T(A059896(i,j), k) = A059896(T(i,k), T(j,k)) (T distributes over A059896).
%F A306697 T(A019565(i), 2^j) = A019565(i)^j.
%F A306697 T(A225546(i), A225546(j)) = A225546(T(i,j)).
%F A306697 (End)
%e A306697 Array T(n, k) begins:
%e A306697 n\k| 1 2 3 4 5 6 7 8 9 10 11 12
%e A306697 ---+-------------------------------------------------------------
%e A306697 1| 1 1 1 1 1 1 1 1 1 1 1 1
%e A306697 2| 1 2 3 4 5 6 7 8 9 10 11 12
%e A306697 3| 1 3 5 9 7 15 11 27 25 21 13 45
%e A306697 4| 1 4 9 16 25 36 49 64 81 100 121 144
%e A306697 5| 1 5 7 25 11 35 13 125 49 55 17 175
%e A306697 6| 1 6 15 36 35 30 77 216 225 210 143 540
%e A306697 7| 1 7 11 49 13 77 17 343 121 91 19 539
%e A306697 8| 1 8 27 64 125 216 343 128 729 1000 1331 1728
%e A306697 9| 1 9 25 81 49 225 121 729 625 441 169 2025
%e A306697 10| 1 10 21 100 55 210 91 1000 441 110 187 2100
%e A306697 11| 1 11 13 121 17 143 19 1331 169 187 23 1573
%e A306697 12| 1 12 45 144 175 540 539 1728 2025 2100 1573 720
%o A306697 (PARI) \\ See Links section.
%Y A306697 Columns (some differing for term 1) and equivalently rows: A003961(3), A000290(4), A045966(5), A045968(7), A045970(11).
%Y A306697 Cf. A000040, A001221, A019565, A064547, A225546, A329050.
%Y A306697 Related binary operations: A067138, A059896, A297845/A003991, A329329/A059897.
%K A306697 nonn,tabl
%O A306697 1,5
%A A306697 _Rémy Sigrist_, Mar 05 2019