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A127170 Triangle read by rows: T(n,k) is the number of divisors of n that are divisible by k, with 1 <= k <= n.

%I A127170 #62 Dec 18 2022 16:54:44
%S A127170 1,2,1,2,0,1,3,2,0,1,2,0,0,0,1,4,2,2,0,0,1,2,0,0,0,0,0,1,4,3,0,2,0,0,
%T A127170 0,1,3,0,2,0,0,0,0,0,1,4,2,0,0,2,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,1,6,4,
%U A127170 3,2,0,2,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,0,0,1,4,2,0,0,0,0,2,0,0,0,0,0,0,1
%N A127170 Triangle read by rows: T(n,k) is the number of divisors of n that are divisible by k, with 1 <= k <= n.
%C A127170 Column k lists the terms of A000005 interleaved with k - 1 zeros.
%C A127170 Eigensequence of the triangle = A007557; i.e., sequence A007557 shifts to the left upon multiplication by A127170. - _Gary W. Adamson_, Apr 27 2009
%H A127170 Michael De Vlieger, <a href="/A127170/b127170.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)
%F A127170 A007429(n) = Sum_{i=1..n} i*a(i).
%F A127170 T(n,k) = A000005(n/k), if k divides n, otherwise 0, with n >= 1 and 1 <= k <= n. - _Omar E. Pol_, Apr 01 2015
%e A127170 First 10 rows of the triangle:
%e A127170   1;
%e A127170   2, 1;
%e A127170   2, 0, 1;
%e A127170   3, 2, 0, 1;
%e A127170   2, 0, 0, 0, 1;
%e A127170   4, 2, 2, 0, 0, 1;
%e A127170   2, 0, 0, 0, 0, 0, 1;
%e A127170   4, 3, 0, 2, 0, 0, 0, 1;
%e A127170   3, 0, 2, 0, 0, 0, 0, 0, 1;
%e A127170   4, 2, 0, 0, 2, 0, 0, 0, 0, 1;
%e A127170   ...
%p A127170 T:= (n, k)-> `if`(irem(n, k)=0, numtheory[tau](n/k), 0):
%p A127170 seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Feb 16 2022
%t A127170 Table[Function[D, Table[Count[D, _?(Mod[#, k] == 0 &)], {k, n}]]@ Divisors[n], {n, 12}] // Flatten (* _Michael De Vlieger_, Feb 16 2022 *)
%o A127170 (PARI) tabl(nn) = {m = matrix(nn, nn, n, k, if ((n % k) == 0, 1, 0)); m = m^2; for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ _Michel Marcus_, Apr 01 2015
%Y A127170 Row sums give A007425.
%Y A127170 Cf. A000005, A007557, A051731, A126988, A007429, A127170, A195050.
%K A127170 nonn,tabl
%O A127170 1,2
%A A127170 _Gary W. Adamson_, Jan 06 2007
%E A127170 8 terms taken from Example section and then corrected in Data section by _Omar E. Pol_, Mar 30 2015
%E A127170 Extended beyond a(21) by _Omar E. Pol_, Apr 01 2015
%E A127170 New name (which was a comment dated Mar 30 2015) from _Omar E. Pol_, Feb 16 2022