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 A305720 #22 Oct 21 2020 23:05:23
%S A305720 1,1,1,1,2,1,1,1,1,1,1,4,3,4,1,1,1,1,1,1,1,1,2,1,16,1,2,1,1,1,3,1,1,3,
%T A305720 1,1,1,8,1,4,5,4,1,8,1,1,1,1,1,1,1,1,1,1,1,1,2,9,64,1,6,1,64,9,2,1,1,
%U A305720 1,1,1,1,1,1,1,1,1,1,1,1,4,1,4,1,8,7,8
%N A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.
%C A305720 The array T is completely multiplicative in both parameters.
%C A305720 For any n > 0 and prime number p, T(n, p) is the highest power of p dividing n.
%C A305720 For any function f associating a nonnegative value to any pair of nonnegative values and such that f(0, 0) = 0, we can build an analog of this sequence, say P_f, such that for any prime number p and any n and k > 0 with p-adic valuations i and j, the p-adic valuation of P_f(n, k) equals f(i, j):
%C A305720 f(i, j) P_f
%C A305720 ------- ---
%C A305720 i * j T (this sequence)
%C A305720 i + j A003991 (product)
%C A305720 abs(i-j) A089913
%C A305720 min(i, j) A003989 (GCD)
%C A305720 max(i, j) A003990 (LCM)
%C A305720 i AND j A059895
%C A305720 i OR j A059896
%C A305720 i XOR j A059897
%C A305720 If log(N) denotes the set {log(n) : n is in N, the set of the positive integers}, one can define a binary operation on log(N): with prime factorizations n = Product p_i^e_i and k = Product p_i^f_i, set log(n) o log(k) = Sum_{i} (e_i*f_i) * log(p_i). o has the premises of a scalar product even if log(N) isn't a vector space. T(n, k) can be viewed as exp(log(n) o log(k)). - _Luc Rousseau_, Oct 11 2020
%F A305720 T(n, k) = T(k, n) (T is commutative).
%F A305720 T(m, T(n, k)) = T(T(m, n), k) (T is associative).
%F A305720 T(n, k) = 1 iff gcd(n, k) = 1.
%F A305720 T(n, n) = A054496(n).
%F A305720 T(n, A007947(n)) = n.
%F A305720 T(n, 1) = 1.
%F A305720 T(n, 2) = A006519(n).
%F A305720 T(n, 3) = A038500(n).
%F A305720 T(n, 4) = A006519(n)^2.
%F A305720 T(n, 5) = A060904(n).
%F A305720 T(n, 6) = A065331(n).
%F A305720 T(n, 7) = A268354(n).
%F A305720 T(n, 8) = A006519(n)^3.
%F A305720 T(n, 9) = A038500(n)^2.
%F A305720 T(n, 10) = A132741(n).
%F A305720 T(n, 11) = A268357(n).
%e A305720 Array T(n, k) begins:
%e A305720 n\k| 1 2 3 4 5 6 7 8 9 10
%e A305720 ---+--------------------------------------------------
%e A305720 1| 1 1 1 1 1 1 1 1 1 1
%e A305720 2| 1 2 1 4 1 2 1 8 1 2 -> A006519
%e A305720 3| 1 1 3 1 1 3 1 1 9 1 -> A038500
%e A305720 4| 1 4 1 16 1 4 1 64 1 4
%e A305720 5| 1 1 1 1 5 1 1 1 1 5 -> A060904
%e A305720 6| 1 2 3 4 1 6 1 8 9 2 -> A065331
%e A305720 7| 1 1 1 1 1 1 7 1 1 1 -> A268354
%e A305720 8| 1 8 1 64 1 8 1 512 1 8
%e A305720 9| 1 1 9 1 1 9 1 1 81 1
%e A305720 10| 1 2 1 4 5 2 1 8 1 10 -> A132741
%t A305720 T[n_, k_] := With[{p = FactorInteger[GCD[n, k]][[All, 1]]}, If[p == {1}, 1, Times @@ (p^(IntegerExponent[n, p] * IntegerExponent[k, p]))]];
%t A305720 Table[T[n-k+1, k], {n, 1, 15}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 11 2018 *)
%o A305720 (PARI) T(n, k) = my (p=factor(gcd(n, k))[,1]); prod(i=1, #p, p[i]^(valuation(n, p[i]) * valuation(k, p[i])))
%Y A305720 Cf. A003989, A003990, A003991, A006519, A007947, A038500, A054496, A059895, A059896, A059897, A060904, A065331, A089913, A132741, A268354, A268357.
%K A305720 nonn,tabl,mult
%O A305720 1,5
%A A305720 _Rémy Sigrist_, Jun 09 2018