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 A326376 #14 Jul 06 2019 02:20:13 %S A326376 1,2,1,1,2,1,4,2,2,1,1,4,3,2,1,2,2,4,4,2,1,1,4,5,4,5,2,1,8,2,6,16,4,6, %T A326376 2,1,1,8,7,8,11,4,7,2,1,2,4,8,256,10,90,4,8,2,1,1,4,9,8,17,12,17,4,9, %U A326376 2,1,4,2,10,16,8,47250,14,512,4,10,2,1,1,8 %N A326376 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: for any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) o f(k)) (where o denotes function composition). %C A326376 This sequence has connections with A297845. %C A326376 The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients. %F A326376 For any m, n, k > 0 and any i >= 0: %F A326376 - T(1, k) = 1, %F A326376 - T(2^i, k) = 2^i, %F A326376 - T(3, k) = k, %F A326376 - T(3^i, k) = k^i, %F A326376 - T(5, k) = A297473(k), %F A326376 - T(6, k) = 2*k, %F A326376 - T(n, 1) = A006519(n), %F A326376 - T(n, 2) = A061142(n), %F A326376 - T(n, 3) = n, %F A326376 - T(n, 5) = A319525(n), %F A326376 - T(m*n, k) = T(m, k) * T(n, k). %e A326376 Array T(n, k) begins: %e A326376 n\k| 1 2 3 4 5 6 7 8 9 10 %e A326376 ---+----------------------------------------------------------- %e A326376 1| 1 1 1 1 1 1 1 1 1 1 %e A326376 2| 2 2 2 2 2 2 2 2 2 2 %e A326376 3| 1 2 3 4 5 6 7 8 9 10 %e A326376 4| 4 4 4 4 4 4 4 4 4 4 %e A326376 5| 1 2 5 16 11 90 17 512 625 550 %e A326376 6| 2 4 6 8 10 12 14 16 18 20 %e A326376 7| 1 2 7 256 17 47250 29 134217728 5764801 5656750 %e A326376 8| 8 8 8 8 8 8 8 8 8 8 %e A326376 9| 1 4 9 16 25 36 49 64 81 100 %e A326376 10| 2 4 10 32 22 180 34 1024 1250 1100 %e A326376 The corresponding polynomials are: %e A326376 f(n)\f(k)| 0 1 x 2 x^2 x+1 x^3 3 2*x x^2+1 %e A326376 ---------+--------------------------------------------------------------------- %e A326376 0| 0 0 0 0 0 0 0 0 0 0 %e A326376 1| 1 1 1 1 1 1 1 1 1 1 %e A326376 x| 0 1 x 2 x^2 x+1 x^3 3 2*x x^2+1 %e A326376 2| 2 2 2 2 2 2 2 2 2 2 %e A326376 x^2| 0 1 x^2 4 x^4 x^2+2*x+1 x^6 9 4*x^2 x^4+2*x^2+1 %e A326376 x+1| 1 2 x+1 3 x^2+1 x+2 x^3+1 4 2*x+1 x^2+2 %e A326376 x^3| 0 1 x^3 8 x^6 x^3+3*x^2+3*x+1 x^9 27 8*x^3 x^6+3*x^4+3*x^2+1 %e A326376 3| 3 3 3 3 3 3 3 3 3 3 %e A326376 2*x| 0 2 2*x 4 2*x^2 2*x+2 2*x^3 6 4*x 2*x^2+2 %e A326376 x^2+1| 1 2 x^2+1 5 x^4+1 x^2+2*x+2 x^6+1 10 4*x^2+1 x^4+2*x^2+2 %o A326376 (PARI) g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1)) %o A326376 f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i,2] * v^(primepi(f[i,1]) - 1)) %o A326376 T(n,k) = g(f(n, f(k))) %Y A326376 See A326377 for the main diagonal of T. %Y A326376 Cf. A006519, A061142, A297473, A297845, A319525. %K A326376 nonn,tabl %O A326376 1,2 %A A326376 _Rémy Sigrist_, Jul 02 2019