Original entry on oeis.org
1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1
Offset: 1
Triangle begins
1;
1,1;
1,0,1;
1,0,1,1;
1,0,0,0,1;
1,0,0,1,1,1;
1,0,0,0,0,0,1;
1,0,0,0,1,0,1,1;
A243987
Triangle read by rows: T(n, k) is the number of divisors of n that are less than or equal to k for 1 <= k <= n.
Original entry on oeis.org
1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, 1, 1, 1, 2, 1, 2, 3, 3, 3, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 4, 1, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4
Offset: 1
T(6,4)=3 since there are 3 divisors of 6 that are less than or equal to 4, namely, 1, 2 and 3.
T(n,k) as a triangle, n=1..15:
1,
1, 2,
1, 1, 2,
1, 2, 2, 3,
1, 1, 1, 1, 2,
1, 2, 3, 3, 3, 4,
1, 1, 1, 1, 1, 1, 2,
1, 2, 2, 3, 3, 3, 3, 4,
1, 1, 2, 2, 2, 2, 2, 2, 3,
1, 2, 2, 2, 3, 3, 3, 3, 3, 4
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,
1, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2,
1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4,
1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4
-
a243987 n k = a243987_tabl !! (n-1) !! (k-1)
a243987_row n = a243987_tabl !! (n-1)
a243987_tabl = map (scanl1 (+)) a051731_tabl
-- Reinhard Zumkeller, Apr 22 2015
-
T:=(n,k)->1/n!*eval(diff(sum(x^j/(1-x^j),j=1..k),x$n),x=0):
seq(seq(T(n,k), k=1..n), n=1..10);
# Alternative:
IversonBrackets := expr -> subs(true=1, false=0, evalb(expr)):
T := (n, k) -> add(IversonBrackets(irem(n, j) = 0), j = 1..k):
for n from 1 to 19 do seq(T(n, k), k = 1..n) od; # Peter Luschny, Jan 02 2021
-
T(n, k) = sumdiv(n, d, d<=k); \\ Michel Marcus, Jun 17 2014
A130026
Triangle (n,k) by columns, arithmetic sequences interspersed with k zeros.
Original entry on oeis.org
1, 2, 1, 3, 0, 1, 4, 3, 0, 1, 5, 0, 0, 0, 1, 6, 5, 4, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 8, 7, 0, 5, 0, 0, 0, 1, 9, 0, 7, 0, 0, 0, 0, 0, 1, 10, 9, 0, 0, 6, 0, 0, 0, 0, 1
Offset: 0
First few rows of the triangle:
1;
2, 1;
3, 0, 1;
4, 3, 0, 1;
5, 0, 0, 0, 1;
6, 5, 4, 0, 0, 1;
7, 0, 0, 0, 0, 0, 1;
...
A343803
a(n) = Sum_{k=1..n} k * (number of divisors of n <= k).
Original entry on oeis.org
1, 5, 9, 23, 20, 65, 35, 109, 96, 164, 77, 377, 104, 307, 362, 525, 170, 818, 209, 1008, 690, 725, 299, 2005, 665, 1000, 1122, 1939, 464, 3106, 527, 2517, 1658, 1682, 1894, 5084, 740, 2089, 2298, 5500, 902, 6022, 989, 4701, 5066, 3035, 1175, 10117, 2478, 6069, 3890, 6532, 1484
Offset: 1
a(4) = 1*(1) + 2*(2) + 3*(2) + 4*(3) = 23, i.e.,
(1 times the number of divisors of 4 that are less than or equal to 1)
+ (2 times the number of divisors of 4 that are less than or equal to 2)
+ (3 times the number of divisors of 4 that are less than or equal to 3)
+ (4 times the number of divisors of 4 that are less than or equal to 4).
-
Table[Sum[Sum[k (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 60}]
-
a(n) = my(d=divisors(n)); sum(k=1, n, k*#select(x->(x<=k), d)); \\ Michel Marcus, Apr 30 2021
Showing 1-4 of 4 results.
Comments