A022825 -id:A022825 - cp's OEIS frontend

A351621 a(1) = 1; a(n) = 1 + a(n-1) + Sum_{k=2..n} a(floor(n/k)).

1, 3, 6, 12, 19, 32, 46, 69, 96, 133, 171, 234, 298, 379, 471, 595, 720, 891, 1063, 1288, 1531, 1815, 2100, 2496, 2900, 3371, 3873, 4479, 5086, 5848, 6611, 7530, 8491, 9580, 10691, 12088, 13486, 15059, 16700, 18642, 20585, 22885, 25186, 27818, 30580, 33630, 36681, 40363, 44060, 48208

Offset: 1

Ilya Gutkovskiy, Feb 20 2022

nonn

Partial sums of A345139.

Cf. A001906, A022825, A025523, A078346, A102378, A345139, A351620.

a[1] = 1; a[n_] := a[n] = 1 + a[n - 1] + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 1, 50}]

G.f. A(x) satisfies: A(x) = ( x + Sum_{k>=2} (1 - x^k) * A(x^k) ) / (1 - x)^2.

A359399 a(1) = 1; a(n) = Sum_{k=2..n} k * a(floor(n/k)).

Original entry on oeis.org

1, 2, 5, 11, 16, 31, 38, 62, 80, 105, 116, 194, 207, 242, 287, 383, 400, 526, 545, 675, 738, 793, 816, 1200, 1250, 1315, 1423, 1605, 1634, 1979, 2010, 2394, 2493, 2578, 2683, 3475, 3512, 3607, 3724, 4364, 4405, 4888, 4931, 5217, 5577, 5692, 5739, 7563, 7661, 8011

Offset: 1

Seiichi Manyama, Mar 31 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A022825.

from functools import lru_cache
@lru_cache(maxsize=None)
def A359399(n):
    if n <= 1:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2*(j2-1)-j*(j-1)>>1)*A359399(k1)
        j, k1 = j2, n//j2
    return c+(n*(n+1)-(j-1)*j>>1) # Chai Wah Wu, Mar 31 2023

G.f. A(x) satisfies A(x) = x + (1/(1 - x)) * Sum_{k>=2} k * (1 - x^k) * A(x^k).

A089646 a(n) = Sum(a(floor(n/p)): p prime and p<=n); a(1) = 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 13, 13, 14, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 32, 33, 33, 35, 36, 38, 41, 42, 43, 45, 46, 47, 51, 52, 53, 56, 57, 58, 59, 60, 62, 64, 65, 66, 69, 71, 72, 74, 75, 76, 82, 83, 84, 87, 87, 89, 93, 94, 95, 97, 101, 102, 106

Offset: 1

Reinhard Zumkeller, Jan 02 2004

nonn

			a(10) = a([10/2])+a([10/3])+a([10/5])+a([10/7]) =
a(5)+a(3)+a(2)+a(1) = (a([5/2])+a([5/3])+a([5/5])) + (a([3/2])+a([3/3])) +
a([2/2]) + a(1) = (a(2)+a(1)+a(1)) + (a(1)+a(1)) + a(1) + a(1) = a([2/2]) +
6*a(1) = a(1) + 6*1 = 7.

Cf. A022825.

A332800 Number of permutations sigma of [n] such that (sigma(k) mod sigma(k+1)) <= (sigma(k+1) mod sigma(k+2)) for 1 <= k <= n - 2.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 44, 109, 241, 530, 1176, 3180, 6456, 14835, 34672, 81877, 179434, 479275, 977224, 2503363, 5339049, 11207391, 28379591, 82473713, 166689486, 370775384, 877910547, 2150475950, 4608590865, 12146671367, 24620749285, 64137229920, 143062854926

Offset: 0

Seiichi Manyama, Feb 27 2020

nonn

Conjecture: Number of permutations sigma such that (sigma(k) mod sigma(k+1)) < (sigma(k+1) mod sigma(k+2)) for 1 <= k <= n - 2 is equal to A022825(n). This is true for n <= 19.

			b(n) = sigma(n) mod sigma(n+1).
In case of n = 3.
    |           | b(1),b(2)
----+-----------+----------
  1 | [1, 2, 3] | [1, 2] *
  2 | [1, 3, 2] | [1, 1]
  3 | [2, 1, 3] | [0, 1] *
  4 | [3, 1, 2] | [0, 1] *
In case of n = 4.
    |              | b(1),b(2),b(3)
----+--------------+---------------
  1 | [1, 2, 3, 4] | [1, 2, 3] *
  2 | [1, 3, 2, 4] | [1, 1, 2]
  3 | [1, 4, 3, 2] | [1, 1, 1]
  4 | [2, 1, 3, 4] | [0, 1, 3] *
  5 | [2, 1, 4, 3] | [0, 1, 1]
  6 | [3, 1, 2, 4] | [0, 1, 2] *
  7 | [4, 1, 2, 3] | [0, 1, 2] *
  8 | [4, 1, 3, 2] | [0, 1, 1]
  9 | [4, 2, 1, 3] | [0, 0, 1]
* (strongly increasing)

Cf. A022825.

a(17)-a(20) from Alois P. Heinz, Feb 27 2020
a(21)-a(22) from Giovanni Resta, Mar 03 2020
a(23)-a(31) from Bert Dobbelaere, Mar 12 2020
a(32) from Bert Dobbelaere, Mar 15 2020

cp's OEIS Frontend

A351621 a(1) = 1; a(n) = 1 + a(n-1) + Sum_{k=2..n} a(floor(n/k)).

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A359399 a(1) = 1; a(n) = Sum_{k=2..n} k * a(floor(n/k)).

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A089646 a(n) = Sum(a(floor(n/p)): p prime and p<=n); a(1) = 1.

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A332800 Number of permutations sigma of [n] such that (sigma(k) mod sigma(k+1)) <= (sigma(k+1) mod sigma(k+2)) for 1 <= k <= n - 2.

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