A013936 -id:A013936 - cp's OEIS frontend

A306533 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} floor(n/j^k).

1, 1, 4, 1, 3, 9, 1, 2, 5, 16, 1, 2, 3, 8, 25, 1, 2, 3, 5, 10, 36, 1, 2, 3, 4, 6, 14, 49, 1, 2, 3, 4, 5, 7, 16, 64, 1, 2, 3, 4, 5, 6, 8, 20, 81, 1, 2, 3, 4, 5, 6, 7, 10, 23, 100, 1, 2, 3, 4, 5, 6, 7, 9, 12, 27, 121, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 29, 144, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 35, 169

Offset: 1

Ilya Gutkovskiy, Feb 22 2019

			Square array begins:
   1,   1,  1,  1,  1,  1,  ...
   4,   3,  2,  2,  2,  2,  ...
   9,   5,  3,  3,  3,  3,  ...
  16,   8,  5,  4,  4,  4,  ...
  25,  10,  6,  5,  5,  5,  ...
  36,  14,  7,  6,  6,  6,  ...

Columns k=0..4 give A000290, A006218, A013936, A013937, A013938.

Cf. A306534.

Table[Function[k, Sum[Floor[n/j^k], {j, 1, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

G.f. of column k (for k > 0): (1/(1 - x)) * Sum_{j>=1} x^(j^k)/(1 - x^(j^k)).

A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 10, 11, 13, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 32, 35, 36, 37, 40, 41, 42, 44, 46, 47, 52, 53, 54, 56, 57, 58, 62, 63, 64, 66, 68, 69, 73, 74, 75, 79, 80, 81, 84, 85, 87, 89, 90, 91, 94, 96, 98, 100, 101, 102, 107, 108, 109, 112, 113, 114, 118

Offset: 0

Henry Bottomley, Mar 25 2002

nonn

The summation has floor(1/2 + sqrt(2*n)) = A002024(n) nonzero terms. - Enrique Pérez Herrero, Apr 05 2010

			a(11) = floor(11/1) + floor(11/3) + floor(11/6) + floor(11/10) + floor(11/15) + ... = 11 + 3 + 1 + 1 + 0 + ... = 16.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000217, A006218, A007862, A013936, A013937, A013938, A013939.

Cf. A002024, A004275. - Enrique Pérez Herrero, Apr 05 2010

[(&+[Floor(n/(k*(k+1)/2)): k in [1..100]]): n in [0..30]]; // G. C. Greubel, May 23 2018

A069470[n_]:=Sum[Floor[(2*n)/(k*(1 + k))], {k, 1, Floor[1/2 + Sqrt[2*n]]}] (* Enrique Pérez Herrero, Apr 05 2010 *)

for(n=0, 30, print1(sum(k=1, 100, floor(n/(k*(k+1)/2))), ", ")) \\ G. C. Greubel, May 23 2018

a(n) = a(n-1) + A007862(n).
It appears that limit((sum(floor((1/2)*n/(k*(k+1))), k=1..n))/n, n=infinity) = 1/2. - Stephen Crowley, Aug 12 2009
From Enrique Pérez Herrero, Apr 05 2010: (Start)
a(n) <= floor((2*n^2)/(1 + n)) = A004275(n).
a(n) <= floor((2*n*floor((1 + 2*sqrt(2*n))/2))/(1+floor((1+2*sqrt(2*n))/2))). (End)
G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^(k*(k+1)/2)). - Ilya Gutkovskiy, Jul 11 2019

A344514 a(n) = Sum_{k=1..n} floor(n/k^2)^k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 12, 14, 15, 16, 22, 23, 24, 25, 34, 35, 43, 44, 54, 55, 56, 57, 69, 71, 72, 92, 106, 107, 108, 109, 140, 141, 142, 143, 199, 200, 201, 202, 222, 223, 224, 225, 247, 309, 310, 311, 400, 402, 434, 435, 461, 462, 554, 555, 583, 584, 585, 586, 616, 617

Offset: 1

Wesley Ivan Hurt, May 21 2021

nonn

			a(9) = Sum_{k=1..9} floor(9/k^2)^k = 9^1 + 2^2 + 1^3 = 14.

Cf. A013936.

Table[Sum[Floor[n/k^2]^k, {k, n}], {n, 100}]

A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107

Offset: 0

Akiva Weinberger, Jan 23 2025

nonn

Partial sum of A060832 except for the first term in the sum.
Congruent to A034968(n) mod 2. Therefore, the parity of a(n) is the parity of the n-th permutation of k elements (k>=n) in lexicographic order.
For even n, a(n) equals A059563(n/2) whenever cosh(1)*n - a(n) < 1. The first time this fails is n=70, as a(70)=107 but A059563(35)=108. For small n, such failures appear to be very rare; however, the asymptotic density of these failures approaches 1.

Cf. A060832, A034968, A013936, A013939, A038663.

a(n) = round(sumpos(k=0, n\(2*k)!)); \\ Michel Marcus, Jan 24 2025

a(n) = cosh(1)*n - f(n) where f(n) = Sum_{k>=0} fract(n/(2k)!). Here, fract() is the fractional part. The error term f(n) is unbounded above, and the greatest lower bound is 0 (even excluding n=0). The first values for which f(n) > s for s=1,2,3 are f(13)=1.06005, f(407) = 2.03382, and f(22319) = 3.01669. The error is almost periodic: for large m, f(n) is approximately f(n+(2m)!). If n is odd, f(n) > 1/2. f(n) alternately rises and descends, that is, f(2*n)f(2*n+2) for all n.

cp's OEIS Frontend

A306533 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} floor(n/j^k).

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A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).

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A344514 a(n) = Sum_{k=1..n} floor(n/k^2)^k.

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A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).

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