Displaying 1-10 of 12 results found.
Number of partitions of n minus number of divisors of n.
+10
23
0, 0, 1, 2, 5, 7, 13, 18, 27, 38, 54, 71, 99, 131, 172, 226, 295, 379, 488, 621, 788, 998, 1253, 1567, 1955, 2432, 3006, 3712, 4563, 5596, 6840, 8343, 10139, 12306, 14879, 17968, 21635, 26011, 31181, 37330, 44581, 53166, 63259, 75169, 89128, 105554, 124752
COMMENTS
a(n) is also the number of partitions of n with at least one distinct part (i.e., not all parts are equal).
MAPLE
with(numtheory): b:= proc(n) option remember; `if`(n=0, 1, add(add(d, d=divisors(j)) *b(n-j), j=1..n)/n) end: a:= n-> b(n)- tau(n):
MATHEMATICA
Table[PartitionsP[n]-DivisorSigma[0, n], {n, 50}] (* Harvey P. Dale, Apr 10 2014 *)
PROG
(PARI) al(n)=vector(n, k, numbpart(k)-numdiv(k))
(Python)
from sympy import npartitions, divisor_count
Number of partitions with maximum rectangle <= n.
+10
4
1, 2, 5, 10, 26, 42, 118, 171, 389, 692, 1442, 1854, 5534, 6895, 11910, 21116, 44278, 52568, 118734, 138670, 300326, 492507, 728514, 829244, 2167430, 2987124, 4167602, 6092588, 11308432, 12554900, 29925267, 33023589, 57950313, 81424281, 106214784, 148101088
COMMENTS
A partition has maximum rectangle <= n iff it is a subpartition of row n of A010766.
FORMULA
a(n) = subpart([<floor(n/k)]). The subpart function is A115728 (or A115729), [<floor(n/k)>] is row n of A010766.
EXAMPLE
The 10 partitions with maximum rectangle <= 3: 0: []; 1: [1]; 2: [2], [1^2], [2,1]; 3: [3], [1^3], [3,1], [2,1^2], [3,1^2].
Total area of the largest inscribed rectangles of all integer partitions of n.
+10
2
0, 1, 4, 8, 18, 29, 54, 82, 136, 202, 309, 441, 658, 915, 1303, 1790, 2479, 3337, 4541, 6022, 8045, 10554, 13876, 17996, 23409, 30055, 38634, 49208, 62650, 79116, 99898, 125213, 156848, 195339, 242964, 300707, 371770, 457493, 562292, 688451, 841707, 1025484
COMMENTS
a(n) >= A000041(n)* A061017(n) for n>0 because the least largest inscribed rectangle of any integer partition of n is A061017(n) and A000041(n) is the number of partitions of n. a(n) >= A116503(n), the sum of the areas of the Durfee squares of all partitions of n.
FORMULA
a(n) = Sum_{k=1..n} k * A115723(n,k) for n>0, a(0) = 0.
EXAMPLE
a(4) = 18 = 4+3+4+3+4 because the partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and the largest inscribed rectangles have areas 4*1, 3*1, 2*2, 1*3, 1*4.
a(5) = 29 = 5+4+4+3+4+4+5 because the partitions of 5 are [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
MAPLE
b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
`if`(i=1, `if`(t+n>k, 0, 1), `if`(i<1, 0, b(n, i-1, t, k)
+add(`if`(t+j>k/i, 0, b(n-i*j, i-1, t+j, k)), j=1..n/i))))
end:
a:= n-> add(k*(b(n, n, 0, k) -b(n, n, 0, k-1)), k=1..n):
seq(a(n), n=0..50);
MATHEMATICA
b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i == 1, If[t + n > k, 0, 1], If[i < 1, 0, b[n, i - 1, t, k] + Sum[If[t + j > k/i, 0, b[n - i j, i - 1, t + j, k]], {j, 1, n/i}]]]];
a[n_] := Sum[k(b[n, n, 0, k] - b[n, n, 0, k - 1]), {k, 1, n}];
Number of partitions of n+2 with largest inscribed rectangle having area <= n.
+10
2
0, 0, 0, 1, 5, 7, 16, 21, 34, 48, 69, 89, 129, 166, 220, 287, 377, 478, 619, 778, 992, 1247, 1565, 1941, 2428, 3000, 3706, 4553, 5594, 6826, 8341, 10129, 12300, 14873, 17962, 21619, 26009, 31175, 37324, 44567, 53164, 63245, 75167, 89118, 105544, 124746, 147261
Number of partitions of n+3 with largest inscribed rectangle having area <= n.
+10
2
0, 0, 0, 0, 3, 5, 17, 22, 39, 57, 85, 107, 162, 208, 273, 360, 474, 597, 774, 970, 1233, 1553, 1937, 2396, 2991, 3694, 4539, 5572, 6822, 8309, 10125, 12278, 14859, 17950, 21605, 25972, 31171, 37312, 44553, 53132, 63241, 75135, 89114, 105522, 124722, 147249
Number of partitions of n+4 with largest inscribed rectangle having area <= n.
+10
2
0, 0, 0, 0, 1, 3, 16, 21, 43, 65, 101, 125, 202, 253, 336, 444, 591, 736, 964, 1193, 1529, 1917, 2390, 2933, 3678, 4519, 5548, 6782, 8303, 10067, 12272, 14819, 17926, 21585, 25946, 31103, 37306, 44533, 53108, 63181, 75129, 89056, 105516, 124682, 147205, 173480
Number of partitions of n+5 with largest inscribed rectangle having area <= n.
+10
2
0, 0, 0, 0, 0, 1, 13, 18, 43, 69, 115, 140, 243, 302, 402, 539, 726, 896, 1183, 1458, 1873, 2356, 2923, 3572, 4489, 5514, 6738, 8231, 10057, 12164, 14809, 17854, 21541, 25912, 31053, 37180, 44523, 53074, 63137, 75017, 89046, 105408, 124672, 147133, 173396
Number of partitions of n+6 with largest inscribed rectangle having area <= n.
+10
2
0, 0, 0, 0, 0, 0, 8, 13, 41, 71, 126, 152, 288, 352, 475, 644, 882, 1077, 1444, 1759, 2286, 2871, 3558, 4317, 5466, 6686, 8161, 9939, 12150, 14632, 17840, 21423, 25842, 31001, 37096, 44313, 53060, 63085, 74947, 88858, 105394, 124492, 147119, 173278, 203994
Number of partitions of n+7 with largest inscribed rectangle having area <= n.
+10
2
0, 0, 0, 0, 0, 0, 4, 8, 35, 67, 132, 158, 330, 399, 547, 755, 1055, 1276, 1737, 2103, 2757, 3476, 4295, 5184, 6605, 8079, 9823, 11956, 14610, 17544, 21401, 25646, 30885, 37014, 44169, 52707, 63063, 74865, 88742, 105074, 124470, 146816, 173256, 203798, 239540
Number of partitions of n+8 with largest inscribed rectangle having area <= n.
+10
2
0, 0, 0, 0, 0, 0, 1, 4, 27, 59, 132, 158, 371, 443, 619, 871, 1246, 1493, 2073, 2485, 3306, 4175, 5154, 6177, 7955, 9703, 11782, 14312, 17514, 20942, 25616, 30583, 36838, 44049, 52479, 62509, 74835, 88622, 104898, 123964, 146786, 172780, 203768, 239236, 280917
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