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 A087623 #15 Jan 17 2025 09:08:03
%S A087623 0,0,1,0,0,2,0,0,0,2,0,0,0,1,4,0,0,0,0,0,2,0,0,0,0,0,2,6,0,0,0,0,0,1,
%T A087623 0,4,0,0,0,0,0,0,0,2,6,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,2,4,10,0,0,
%U A087623 0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,1,0,2,0,6
%N A087623 Square array A(n,k) = the cardinality of the set {x in [1,k-1] : gcd(x,k)=n}, read by rising antidiagonals.
%C A087623 Triangle read by rows: T(m,n) is the cardinality of the set {k in [1,n-1] : gcd(k,n)=m}. - The original definition.
%C A087623 A generalization of Euler's phi function: the n-th term of topmost row = A000010(n), for n > 1.
%H A087623 Antti Karttunen, <a href="/A087623/b087623.txt">Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array (rows of the triangle)</a>
%e A087623 The top left corner of the array:
%e A087623 n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
%e A087623 ---+------------------------------------------------------------------------
%e A087623 1 | 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6,
%e A087623 2 | 0, 0, 0, 1, 0, 2, 0, 2, 0, 4, 0, 2, 0, 6, 0, 4, 0, 6,
%e A087623 3 | 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 2,
%e A087623 4 | 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0,
%e A087623 5 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0,
%e A087623 6 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2,
%e A087623 7 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
%e A087623 8 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
%e A087623 9 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
%e A087623 etc.
%e A087623 A(1,4) = 2 and A(2,4) = 1 because gcd(1,4)=1, gcd(2,4)=2, gcd(3,4)=1.
%e A087623 A(1,12) = 4, A(2,12) = A(3,12) = A(4,12) = 2, and A(6,12) = 1 because gcd(1,12) = gcd(5,12) = gcd(7,12) = gcd(9,12) = 1, gcd(2,12) = gcd(10,12) = 2, gcd(3,12) = gcd(9,12) = 3, gcd(4,12) = gcd(8,12) = 4 and gcd(6,12) = 6.
%o A087623 (PARI)
%o A087623 up_to = 105;
%o A087623 A087623sq(n, k) = sum(x=1,k-1,gcd(x,k)==n);
%o A087623 A087623list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A087623sq((a-(col-1)), col))); (v); };
%o A087623 v087623 = A087623list(up_to);
%o A087623 A087623(n) = v087623[n]; \\ _Antti Karttunen_, Jan 17 2025
%Y A087623 Cf. A000010.
%Y A087623 Cf. also A054523.
%K A087623 nonn,easy,tabl
%O A087623 1,6
%A A087623 Michele Dondi (bik.mido(AT)tiscalinet.it), Sep 14 2003
%E A087623 Definition rephrased in terms of square array instead of triangular table, and data section extended up to 105 terms by _Antti Karttunen_, Jan 17 2025