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A010766 Triangle read by rows: row n gives the numbers floor(n/k), k = 1..n.

%I A010766 #111 Aug 07 2025 08:35:49
%S A010766 1,2,1,3,1,1,4,2,1,1,5,2,1,1,1,6,3,2,1,1,1,7,3,2,1,1,1,1,8,4,2,2,1,1,
%T A010766 1,1,9,4,3,2,1,1,1,1,1,10,5,3,2,2,1,1,1,1,1,11,5,3,2,2,1,1,1,1,1,1,12,
%U A010766 6,4,3,2,2,1,1,1,1,1,1,13,6,4,3,2,2,1,1,1,1,1,1,1
%N A010766 Triangle read by rows: row n gives the numbers floor(n/k), k = 1..n.
%C A010766 Number of times k occurs as divisor of numbers not greater than n. - _Reinhard Zumkeller_, Mar 19 2004
%C A010766 Viewed as a partition, row n is the smallest partition that contains every partition of n in the usual ordering. - _Franklin T. Adams-Watters_, Mar 11 2006
%C A010766 Row sums = A006218. - _Gary W. Adamson_, Oct 30 2007
%C A010766 A014668 = eigensequence of the triangle. A163313 = A010766 * A014668 (diagonalized) as an infinite lower triangular matrix. - _Gary W. Adamson_, Jul 30 2009
%C A010766 A018805(T(n,k)) = A242114(n,k). - _Reinhard Zumkeller_, May 04 2014
%C A010766 Viewed as partitions, all rows are self-conjugate. - _Matthew Vandermast_, Sep 10 2014
%C A010766 Row n is the partition whose Young diagram is the union of Young diagrams of all partitions of n (rewording of Franklin T. Adams-Watters's comment). - _Harry Richman_, Jan 13 2022
%D A010766 Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
%H A010766 T. D. Noe, <a href="/A010766/b010766.txt">Rows n = 1..50 of triangle, flattened</a>
%F A010766 G.f.: 1/(1-x)*Sum_{k>=1} x^k/(1-y*x^k). - _Vladeta Jovovic_, Feb 05 2004
%F A010766 Triangle A010766 = A000012 * A051731 as infinite lower triangular matrices. - _Gary W. Adamson_, Oct 30 2007
%F A010766 Equals A000012 * A051731 as infinite lower triangular matrices. - _Gary W. Adamson_, Nov 14 2007
%F A010766 Let T(n,0) = n+1, then T(n,k) = (sum of the k preceding elements in the previous column) minus (sum of the k preceding elements in same column). - _Mats Granvik_, _Gary W. Adamson_, Feb 20 2010
%F A010766 T(n,k) = (n - A048158(n,k)) / k. - _Reinhard Zumkeller_, Aug 13 2013
%F A010766 T(n,k) = 1 + T(n-k,k) (where T(n-k,k) = 0 if n < 2*k). - _Robert Israel_, Sep 01 2014
%F A010766 T(n,k) = T(floor(n/k),1) if k>1; T(n,1) = 1 - Sum_{i=2..n} A008683(i)*T(n,i). If we modify the formula to T(n,1) = 1 - Sum_{i=2..n} A008683(i)*T(n,i)/i^s, where s is a complex variable, then the first column becomes the partial sums of the Riemann zeta function. - _Mats Granvik_, Apr 27 2016
%e A010766 Triangle starts:
%e A010766    1:  1;
%e A010766    2:  2,  1;
%e A010766    3:  3,  1, 1;
%e A010766    4:  4,  2, 1, 1;
%e A010766    5:  5,  2, 1, 1, 1;
%e A010766    6:  6,  3, 2, 1, 1, 1;
%e A010766    7:  7,  3, 2, 1, 1, 1, 1;
%e A010766    8:  8,  4, 2, 2, 1, 1, 1, 1;
%e A010766    9:  9,  4, 3, 2, 1, 1, 1, 1, 1;
%e A010766   10: 10,  5, 3, 2, 2, 1, 1, 1, 1, 1;
%e A010766   11: 11,  5, 3, 2, 2, 1, 1, 1, 1, 1, 1;
%e A010766   12: 12,  6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1;
%e A010766   13: 13,  6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1;
%e A010766   14: 14,  7, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
%e A010766   15: 15,  7, 5, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
%e A010766   16: 16,  8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
%e A010766   17: 17,  8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A010766   18: 18,  9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A010766   19: 19,  9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A010766   20: 20, 10, 6, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A010766   ...
%p A010766 seq(seq(floor(n/k),k=1..n),n=1..20); # _Robert Israel_, Sep 01 2014
%t A010766 Flatten[Table[Floor[n/k],{n,20},{k,n}]] (* _Harvey P. Dale_, Nov 03 2012 *)
%o A010766 (Haskell)
%o A010766 a010766 = div
%o A010766 a010766_row n = a010766_tabl !! (n-1)
%o A010766 a010766_tabl = zipWith (map . div) [1..] a002260_tabl
%o A010766 -- _Reinhard Zumkeller_, Apr 29 2015, Aug 13 2013, Apr 13 2012
%o A010766 (PARI) a(n)=t=floor((-1+sqrt(1+8*(n-1)))/2);(t+1)\(n-t*(t+1)/2) \\ _Edward Jiang_, Sep 10 2014
%o A010766 (PARI) T(n, k) = sum(i=1, n, (i % k) == 0); \\ _Michel Marcus_, Apr 08 2017
%Y A010766 Another version of A003988.
%Y A010766 Finite differences of rows: A075993.
%Y A010766 Cf. related triangles: A002260, A013942, A051731, A163313, A277646, A277647.
%Y A010766 Cf. related sequences: A006218, A014668, A115725.
%Y A010766 Columns of this triangle:
%Y A010766 T(n,1) = n,
%Y A010766 T(n,2) = A008619(n-2) for n>1,
%Y A010766 T(n,3) = A008620(n-3) for n>2,
%Y A010766 T(n,4) = A008621(n-4) for n>3,
%Y A010766 T(n,5) = A002266(n) for n>4,
%Y A010766 T(n,n) = A000012(n) = 1.
%Y A010766 Rows of this triangle (with infinite trailing zeros):
%Y A010766 T(1,k) = A000007(k-1),
%Y A010766 T(2,k) = A033322(k),
%Y A010766 T(3,k) = A278105(k),
%Y A010766 T(4,k) = A033324(k),
%Y A010766 T(5,k) = A033325(k),
%Y A010766 T(6,k) = A033326(k),
%Y A010766 T(7,k) = A033327(k),
%Y A010766 T(8,k) = A033328(k),
%Y A010766 T(9,k) = A033329(k),
%Y A010766 T(10,k) = A033330(k),
%Y A010766 ...
%Y A010766 T(99,k) = A033419(k),
%Y A010766 T(100,k) = A033420(k),
%Y A010766 T(1000,k) = A033421(k),
%Y A010766 T(10^4,k) = A033422(k),
%Y A010766 T(10^5,k) = A033427(k),
%Y A010766 T(10^6,k) = A033426(k),
%Y A010766 T(10^7,k) = A033425(k),
%Y A010766 T(10^8,k) = A033424(k),
%Y A010766 T(10^9,k) = A033423(k).
%K A010766 nonn,tabl,easy,nice
%O A010766 1,2
%A A010766 _N. J. A. Sloane_
%E A010766 Cross references edited by _Jason Kimberley_, Nov 23 2016