A144300
Number of partitions of n minus number of divisors of n.
Original entry on oeis.org
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
Offset: 1
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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):
seq(a(n), n=1..50); # Alois P. Heinz, Oct 07 2008
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Table[PartitionsP[n]-DivisorSigma[0,n],{n,50}] (* Harvey P. Dale, Apr 10 2014 *)
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al(n)=vector(n,k,numbpart(k)-numdiv(k))
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from sympy import npartitions, divisor_count
def A144300(n): return npartitions(n)-divisor_count(n) # Chai Wah Wu, Oct 16 2023
A115725
Number of partitions with maximum rectangle <= n.
Original entry on oeis.org
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
Offset: 0
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].
A182099
Total area of the largest inscribed rectangles of all integer partitions of n.
Original entry on oeis.org
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
Offset: 0
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].
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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);
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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}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 06 2020, after Alois P. Heinz *)
A218623
Number of partitions of n+2 with largest inscribed rectangle having area <= n.
Original entry on oeis.org
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
Offset: 0
A218624
Number of partitions of n+3 with largest inscribed rectangle having area <= n.
Original entry on oeis.org
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
Offset: 0
A218625
Number of partitions of n+4 with largest inscribed rectangle having area <= n.
Original entry on oeis.org
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
Offset: 0
A218626
Number of partitions of n+5 with largest inscribed rectangle having area <= n.
Original entry on oeis.org
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
Offset: 0
A218627
Number of partitions of n+6 with largest inscribed rectangle having area <= n.
Original entry on oeis.org
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
Offset: 0
A218628
Number of partitions of n+7 with largest inscribed rectangle having area <= n.
Original entry on oeis.org
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
Offset: 0
A218629
Number of partitions of n+8 with largest inscribed rectangle having area <= n.
Original entry on oeis.org
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
Offset: 0
Showing 1-10 of 12 results.
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