cp's OEIS Frontend

A355474 Square array T(m,n) = Card({ (i, j) : 1 <= i <= m, 1 <= j <= min(n, i), GCD(i, j) = 1 }), read by antidiagonals upwards.

%I A355474 #25 Sep 24 2022 15:46:24
%S A355474 1,2,1,3,2,1,4,4,2,1,5,5,4,2,1,6,7,6,4,2,1,7,8,9,6,4,2,1,8,10,10,10,6,
%T A355474 4,2,1,9,11,13,11,10,6,4,2,1,10,13,15,15,12,10,6,4,2,1,11,14,17,17,17,
%U A355474 12,10,6,4,2,1,12,16,19,20,20,18,12,10,6,4,2,1
%N A355474 Square array T(m,n) = Card({ (i, j) : 1 <= i <= m, 1 <= j <= min(n, i), GCD(i, j) = 1 }), read by antidiagonals upwards.
%C A355474 Also the number of regions in the 0 < x < y sector of the plane that are delimited by the lines with equations i*x + j*y = 0, where i and j are integers, not both 0, and |i| <= m, |j| <= n. This remark is motivated by Factorization Patterns (FPs) and Factorization Patterns of Sequences of Divisors (FPSD) concerns, as defined in A191743 and A290110. This is the case k=2 of a more general problem where k is omega(z)=A001221(z), the number of distinct primes dividing z, for which we would define T(n1,n2,...,nk) instead of T(m,n). The idea is the following: two numbers (e.g., 12 and 20) can have the same FP (p^2*q) without having the same FPSD ([1 < p < q < p^2 < p*q < p^2*q] != [1 < p < p^2 < q < p*q < p^2*q]). T(m,n) tells how many distinct FPSDs share the same FP of the p^m*q^n form. See the illustration for (m,n) = (2,1), section Links.
%H A355474 Luc Rousseau, <a href="/A355474/a355474.png">Illustration of T(2,1)=2 and its relationship with the p^2*q factorization pattern</a>.
%F A355474 T(n,n) = A002088(n).
%e A355474 Let m=2 and n=1. There are exactly two lattice points (i, j) that satisfy 1 <= i <= 2 and 1 <= j <= min(1, i) and GCD(i, j) = 1, namely (1, 1) and (2, 1). So T(2,1) = 2.
%e A355474 Array begins:
%e A355474   m\n|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
%e A355474   ---+----------------------------------------------------
%e A355474    1 |  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
%e A355474    2 |  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
%e A355474    3 |  3  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4
%e A355474    4 |  4  5  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6
%e A355474    5 |  5  7  9 10 10 10 10 10 10 10 10 10 10 10 10 10 10
%e A355474    6 |  6  8 10 11 12 12 12 12 12 12 12 12 12 12 12 12 12
%e A355474    7 |  7 10 13 15 17 18 18 18 18 18 18 18 18 18 18 18 18
%e A355474    8 |  8 11 15 17 20 21 22 22 22 22 22 22 22 22 22 22 22
%e A355474    9 |  9 13 17 20 24 25 27 28 28 28 28 28 28 28 28 28 28
%e A355474   10 | 10 14 19 22 26 27 30 31 32 32 32 32 32 32 32 32 32
%e A355474   11 | 11 16 22 26 31 33 37 39 41 42 42 42 42 42 42 42 42
%e A355474   12 | 12 17 23 27 33 35 40 42 44 45 46 46 46 46 46 46 46
%e A355474   13 | 13 19 26 31 38 41 47 50 53 55 57 58 58 58 58 58 58
%e A355474   14 | 14 20 28 33 41 44 50 53 57 59 62 63 64 64 64 64 64
%e A355474   15 | 15 22 30 36 44 47 54 58 62 64 68 69 71 72 72 72 72
%e A355474   16 | 16 23 32 38 47 50 58 62 67 69 74 75 78 79 80 80 80
%e A355474   17 | 17 25 35 42 52 56 65 70 76 79 85 87 91 93 95 96 96
%o A355474 (PARI)
%o A355474 T(m, n) = sum(i=1, m, sum(j=1, min(n, i), gcd(i, j)==1))
%o A355474 for(d=2,10,for(n=1,d-1,my(m=d-n);print1(T(m,n),", ")))
%Y A355474 Cf. A191743, A290110, A002088, A001221.
%K A355474 nonn,tabl
%O A355474 1,2
%A A355474 _Luc Rousseau_, Jul 03 2022