A024917 -id:A024917 - cp's OEIS frontend

A345682 a(n) = n! * Sum_{k=1..n} 1/(k*floor(n/k)).

1, 2, 7, 26, 148, 804, 6228, 47424, 441936, 4288320, 50437440, 560373120, 7723935360, 106618256640, 1614841401600, 25127582054400, 446784010444800, 7727747269939200, 152873884406476800, 2966599550251008000, 62987912790921216000, 1378192085174919168000

Offset: 1

Vaclav Kotesovec, Jun 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..449
Vaclav Kotesovec, Plot of a(n)/n! for n = 1..1000000

Cf. A006218, A010786, A024917, A092143, A117871, A345683, A345684.

Table[n! * Sum[1/(k*Floor[n/k]), {k, 1, n}], {n, 1, 25}]
Table[n! * Sum[(HarmonicNumber[Floor[n/j]] - HarmonicNumber[Floor[n/(1 + j)]])/j, {j, 1, n}], {n, 1, 25}]

a(n) = n!*sum(k=1, n, 1/(k*(n\k))); \\ Michel Marcus, Jun 24 2021

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (1-x^k)*log(1-x^k)/k)/(1-x))) \\ Seiichi Manyama, Jul 23 2022

a(n) ~ c * n!, where c = Sum_{j>=1} log(1 + 1/j)/j = A131688 = 1.25774...

E.g.f.: -(1/(1-x)) * Sum_{k>0} (1 - x^k) * log(1 - x^k)/k. - Seiichi Manyama, Jul 23 2022

A224526 Number of idempotent 4 X 4 0..n matrices of rank 1.

Original entry on oeis.org

108, 404, 892, 1716, 2732, 4324, 6060, 8516, 11308, 14820, 18572, 23668, 28716, 34916, 41836, 49860, 58076, 68164, 78252, 90356, 102988, 116868, 131276, 148564, 165660, 184532, 204604, 226788, 249116, 274900, 300252, 328628, 357868, 389028, 421580, 457924, 493500

Offset: 1

R. H. Hardin, Apr 09 2013

nonn

Row 4 of A224524

			Some solutions for n=3:
  0 0 0 0     0 0 0 0     1 0 0 0     0 0 0 0     0 0 0 0
  0 0 0 0     2 0 0 1     0 0 0 0     1 1 0 0     0 0 0 0
  2 0 1 0     2 0 0 1     0 0 0 0     3 3 0 0     0 0 0 0
  0 0 0 0     2 0 0 1     0 0 0 0     2 2 0 0     1 0 1 1

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A002541, A024917, A224524.

F4 := k -> 8*k^3 + 36*k^2 + 24*add(m*floor(k/m), m = 2 .. k) + 12*add(floor(k/m), m = 2 .. k) + 12*add(floor(k/m)^2, m = 2 .. k) + 60*k + 4:
map(F4, [$1..100]); # Robert Israel, Dec 15 2019

Table[8*n^3+36*n^2+60*n+4+24*Sum[k*Floor[n/k],{k, 2, n}]+12*Sum[Floor[(n-k)/k],{k, n-1}]+12*Sum[Floor[(n/k)]^2,{k,2,n}],{n,1,100}] (* Metin Sariyar, Dec 15 2019 *)

a(n) = 8*n^3 + 36*n^2 + 60*n + 4 + 24*A024917(n) + 12*A002541(n) + 12*Sum_{m=2..n} floor(n/m)^2. - Robert Israel, Dec 15 2019

More terms from Metin Sariyar, Dec 15 2019

A303900 Numbers k such that the average of all strong divisors of all positive integers <= k is an integer.

Original entry on oeis.org

2, 8, 12, 16, 67, 924122, 1067239

Offset: 1

Ilya Gutkovskiy, May 02 2018

We say d is a strong divisor of k iff d is a divisor of k and d > 1.
Numbers k such that A002541(k) | A024917(k).
a(8) > 10^12. - Giovanni Resta, May 05 2018

Index entries for sequences related to divisors of numbers

Cf. A024917, A027750, A032741, A039653, A056550, A226647, A284288, A284755, A303480, A303482, A303659.

a(3)-a(7) corrected by Daniel Suteu, May 03 2018

cp's OEIS Frontend

A345682 a(n) = n! * Sum_{k=1..n} 1/(k*floor(n/k)).

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A224526 Number of idempotent 4 X 4 0..n matrices of rank 1.

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A303900 Numbers k such that the average of all strong divisors of all positive integers <= k is an integer.

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