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A333292 Triangle read by rows: T(m,n) = Sum_{ 1 <= i <= m, 1 <= j <= n, gcd(i,j)=1 } i*j, for 1 <= n <= m.

%I A333292 #29 Mar 16 2021 19:00:49
%S A333292 1,3,5,6,14,23,10,18,39,55,15,33,69,105,155,21,39,75,111,191,227,28,
%T A333292 60,117,181,296,374,521,36,68,149,213,368,446,649,777,45,95,176,276,
%U A333292 476,554,820,1020,1263,55,105,216,316,516,594,930,1130,1463,1663,66,138,282,426,681,825,1238,1526,1958,2268,2873
%N A333292 Triangle read by rows: T(m,n) = Sum_{ 1 <= i <= m, 1 <= j <= n, gcd(i,j)=1 } i*j, for 1 <= n <= m.
%C A333292 The last two diagonals are A333293, Sum_{k=1..n-1} k^2*phi(k) + n^2*phi(n)/2, and A319087, Sum_{k=1..n} k^2*phi(k), where phi = A000010. Is there a similar formula for the general term?
%H A333292 Alois P. Heinz, <a href="/A333292/b333292.txt">Rows n = 1..141, flattened</a>
%e A333292 Triangle begins:
%e A333292 1,
%e A333292 3, 5,
%e A333292 6, 14, 23,
%e A333292 10, 18, 39, 55,
%e A333292 15, 33, 69, 105, 155,
%e A333292 21, 39, 75, 111, 191, 227,
%e A333292 28, 60, 117, 181, 296, 374, 521,
%e A333292 36, 68, 149, 213, 368, 446, 649, 777,
%e A333292 45, 95, 176, 276, 476, 554, 820, 1020, 1263,
%e A333292 55, 105, 216, 316, 516, 594, 930, 1130, 1463, 1663,
%e A333292 ...
%p A333292 T:= (m, n)-> add(add(`if`(igcd(i, j)=1, i*j, 0), j=1..n), i=1..m):
%p A333292 seq(seq(T(m, n), n=1..m), m=1..12);  # _Alois P. Heinz_, Mar 23 2020
%Y A333292 First two columns are A000217 and A074378, rightmost two diagonals are A333293 and A319087.
%Y A333292 Main diagonal is A319087.
%Y A333292 Cf. A320541.
%K A333292 nonn,tabl
%O A333292 1,2
%A A333292 _N. J. A. Sloane_, Mar 23 2020