Search: a297024 -id:a297024
|
| |
|
|
A296955
|
|
Sum of the smaller parts of the partitions of n into two distinct parts such that the smaller part divides the larger.
|
|
+10
5
|
|
|
|
0, 0, 1, 1, 1, 3, 1, 3, 4, 3, 1, 10, 1, 3, 9, 7, 1, 12, 1, 12, 11, 3, 1, 24, 6, 3, 13, 14, 1, 27, 1, 15, 15, 3, 13, 37, 1, 3, 17, 30, 1, 33, 1, 18, 33, 3, 1, 52, 8, 18, 21, 20, 1, 39, 17, 36, 23, 3, 1, 78, 1, 3, 41, 31, 19, 45, 1, 24, 27, 39, 1, 87, 1, 3, 49, 26, 19, 51, 1, 66, 40, 3, 1, 98, 23, 3, 33, 48, 1, 99, 21, 30, 35, 3, 25, 108, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,6
|
|
|
COMMENTS
|
The number of partitions of n into 3 parts whose "middle" part divides n. - Wesley Ivan Hurt, Oct 21 2021
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{i=1..floor((n-1)/2)} i * (floor(n/i) - floor((n-1)/i).
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+2))*( n*q^(3*n+4) - (n + 1)*q^(2*n+2) - (n - 1)*q^(n+2) + n )/( (1 - q^n )*(1 - q^(n+2))^2 ). (In equation 1 in Arndt, after combining the two n = 0 summands to get t/(1 - t), apply the operator t*d/dt and then set t = q^2 and x = 1. Cf. A001065.) - Peter Bala, Jan 22 2021
|
|
|
EXAMPLE
|
a(12) = 10; the partitions of 12 into two distinct parts are (11,1), (10,2), (9,3), (8,4) and (7,5). 1 divides 11, 2 divides 10, 3 divides 9 and 4 divides 8, so the sum of the smaller parts gives 1 + 2 + 3 + 4 = 10.
|
|
|
MAPLE
|
with(numtheory):
a := n -> add( d, d = divisors(n) minus {floor((n+1)/2), n} ):
|
|
|
MATHEMATICA
|
Table[Sum[i (Floor[n/i] - Floor[(n - 1)/i]), {i, Floor[(n - 1)/2]}], {n, 100}]
f[n_] := Plus @@ Select[Divisors@n, 2 # < n &]; Array[f, 75] (* Robert G. Wilson v, Dec 23 2017 *)
|
|
|
PROG
|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,changed
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A303384
|
|
Total area of all rectangles with dimensions s and t where s | t, n = s + t and s <= t.
|
|
+10
2
|
|
|
|
0, 1, 2, 7, 4, 22, 6, 35, 26, 50, 10, 126, 12, 86, 100, 155, 16, 247, 18, 294, 172, 182, 22, 590, 124, 242, 260, 518, 28, 860, 30, 651, 364, 386, 380, 1365, 36, 470, 484, 1390, 40, 1532, 42, 1134, 1144, 662, 46, 2542, 342, 1395, 772, 1526, 52, 2380, 788
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * (floor((n-i)/i) - floor((n-i-1)/i)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) - zeta(3) = 0.442877... . (End)
|
|
|
MAPLE
|
with(numtheory): seq(n*sigma(n) - sigma[2](n), n=1..60); # Ridouane Oudra, Apr 15 2021
|
|
|
MATHEMATICA
|
Table[Sum[i (n - i) (Floor[(n - i)/i] - Floor[(n - i - 1)/i]), {i, Floor[n/2]}], {n, 80}]
a[n_] := n * DivisorSigma[1, n] - DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Dec 11 2023 *)
|
|
|
PROG
|
(Magma) [0] cat [&+[k*(n-k)*((n-k) div k)-(n-k-1) div k: k in [1..n div 2]]: n in [2..80]]; // Vincenzo Librandi, Jun 07 2018
(PARI) a(n) = sum(i=1, n\2, i*(n-i)*((n-i)\i - (n-i-1)\i)); \\ Michel Marcus, Jun 07 2018
(PARI) a(n) = sumdiv(n, d, d*(n-d)); \\ Daniel Suteu, Jun 19 2018
(PARI) a(n) = {my(f = factor(n)); n * sigma(f) - sigma(f, 2); } \\ Amiram Eldar, Dec 11 2023
(GAP) List([1..60], n->Sum([1..Int(n/2)], i->i*(n-i)*(Int((n-i)/i)-Int((n-i-1)/i)))); # Muniru A Asiru, Jun 07 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A297053
|
|
Sum of the larger parts of the partitions of n into two parts such that the smaller part does not divide the larger.
|
|
+10
0
|
|
|
|
0, 0, 0, 0, 3, 0, 9, 5, 12, 13, 30, 7, 45, 38, 41, 43, 84, 48, 108, 67, 103, 124, 165, 78, 178, 185, 192, 175, 273, 162, 315, 247, 308, 343, 350, 244, 459, 440, 451, 360, 570, 411, 630, 535, 545, 670, 759, 496, 786, 718, 818, 787, 975, 768, 959, 834, 1042
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{i=1..floor(n/2)} (n-i) * (1 - (floor(n/i) - floor((n-1)/i))).
|
|
|
EXAMPLE
|
a(10) = 13; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). The sum of the larger parts of these partitions such that the smaller part does not divide the larger is then 7 + 6 = 13.
|
|
|
MATHEMATICA
|
Table[Sum[(n - i) (1 - (Floor[n/i] - Floor[(n - 1)/i])), {i, Floor[n/2]}], {n, 80}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.005 seconds
|
