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A135539 Triangle read by rows: T(n,k) = number of divisors of n that are >= k.

%I A135539 #35 Jan 08 2024 01:36:51
%S A135539 1,2,1,2,1,1,3,2,1,1,2,1,1,1,1,4,3,2,1,1,1,2,1,1,1,1,1,1,4,3,2,2,1,1,
%T A135539 1,1,3,2,2,1,1,1,1,1,1,4,3,2,2,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,6,5,
%U A135539 4,3,2,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1
%N A135539 Triangle read by rows: T(n,k) = number of divisors of n that are >= k.
%C A135539 Row sums give A000203.
%C A135539 Left border is A000005.
%H A135539 Seiichi Manyama, <a href="/A135539/b135539.txt">Rows n = 1..140, flattened</a>
%F A135539 Triangle read by rows, partial sums of A051731 starting from the right. A051731 as a lower triangular matrix times an all 1's lower triangular matrix.
%F A135539 From _Seiichi Manyama_, Jan 07 2023: (Start)
%F A135539 G.f. of column k: Sum_{j>=1} x^(k*j)/(1 - x^j).
%F A135539 G.f. of column k: Sum_{j>=k} x^j/(1 - x^j). (End)
%F A135539 Sum_{j=1..n} T(j, k) ~ n * (log(n) + 2*gamma - 1 - H(k-1)), where gamma is Euler's constant (A001620), and H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - _Amiram Eldar_, Jan 08 2024
%e A135539 First few rows of the triangle:
%e A135539   1;
%e A135539   2, 1;
%e A135539   2, 1, 1;
%e A135539   3, 2, 1, 1;
%e A135539   2, 1, 1, 1, 1;
%e A135539   4, 3, 2, 1, 1, 1;
%e A135539   2, 1, 1, 1, 1, 1, 1;
%e A135539   4, 3, 2, 2, 1, 1, 1, 1;
%e A135539   3, 2, 2, 1, 1, 1, 1, 1, 1;
%e A135539   4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
%e A135539   2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A135539   6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
%e A135539   ...
%p A135539 with(numtheory);
%p A135539 f1:=proc(n) local d,s1,t1,t2,i;
%p A135539 d:=tau(n);
%p A135539 s1:=sort(divisors(n));
%p A135539 t1:=Array(1..n,0);
%p A135539 for i from 1 to d do t1[n-s1[i]+1]:=1; od:
%p A135539 t2:=PSUM(convert(t1,list));
%p A135539 [seq(t2[n+1-i],i=1..n)];
%p A135539 end proc;
%p A135539 for n from 1 to 15 do lprint(f1(n)); od: # _N. J. A. Sloane_, Nov 09 2018
%t A135539 T[n_, k_] := DivisorSum[n, Boole[# >= k]&];
%t A135539 Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 15 2023 *)
%o A135539 (PARI) row(n) = my(d=divisors(n)); vector(n, k, #select(x->(x>=k), d)); \\ _Michel Marcus_, Jul 23 2022
%Y A135539 Column k=1..10 give A000005, A032741, A023645, A321014, A338648, A338649, A338650, A338651, A338652, A338653.
%Y A135539 Cf. A051731, A000203.
%Y A135539 Cf. A001008, A001620, A002805.
%K A135539 nonn,easy,tabl
%O A135539 1,2
%A A135539 _Gary W. Adamson_, Oct 30 2007
%E A135539 Clearer definition from _N. J. A. Sloane_, Nov 09 2018