A067538
Number of partitions of n in which the number of parts divides n.
Original entry on oeis.org
1, 2, 2, 4, 2, 8, 2, 11, 9, 14, 2, 46, 2, 24, 51, 66, 2, 126, 2, 202, 144, 69, 2, 632, 194, 116, 381, 756, 2, 1707, 2, 1417, 956, 316, 2043, 5295, 2, 511, 2293, 9151, 2, 10278, 2, 8409, 14671, 1280, 2, 36901, 8035, 21524, 11614, 25639, 2, 53138, 39810, 85004
Offset: 1
a(3)=2 because 3 is a prime; a(4)=4 because the five partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}, and the number of parts in each of them divides 4 except for {2, 1, 1}.
From _Gus Wiseman_, Sep 24 2019: (Start)
The a(1) = 1 through a(8) = 11 partitions whose length divides their sum are the following. The Heinz numbers of these partitions are given by A316413.
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (11111) (33) (1111111) (44)
(31) (42) (53)
(1111) (51) (62)
(222) (71)
(321) (2222)
(411) (3221)
(111111) (3311)
(4211)
(5111)
(11111111)
(End)
Partitions with integer geometric mean are
A067539.
-
Do[p = IntegerPartitions[n]; l = Length[p]; c = 0; k = 1; While[k < l + 1, If[ IntegerQ[ n/Length[ p[[k]] ]], c++ ]; k++ ]; Print[c], {n, 1, 57}, All]
p[n_,k_]:=p[n,k]=p[n-1,k-1]+p[n-k,k];p[n_,k_]:=0/;k>n;p[n_,n_]:=1;p[n_,0]:=0
Table[Plus @@ (p[n,# ]&/ @ Divisors[n]),{n,36}] (* Wouter Meeussen, Jun 07 2009 *)
Table[Count[IntegerPartitions[n], q_ /; IntegerQ[Mean[q]]], {n, 50}] (*Clark Kimberling, Apr 23 2019 *)
-
a(n) = {my(nb = 0); forpart(p=n, if ((vecsum(Vec(p)) % #p) == 0, nb++);); nb;} \\ Michel Marcus, Jul 03 2018
-
# uses A008284_T
from sympy import divisors
def A067538(n): return sum(A008284_T(n,d) for d in divisors(n,generator=True)) # Chai Wah Wu, Sep 21 2023
A102627
Number of partitions of n into distinct parts in which the number of parts divides n.
Original entry on oeis.org
1, 1, 1, 2, 1, 4, 1, 4, 4, 5, 1, 15, 1, 7, 14, 17, 1, 28, 1, 40, 28, 11, 1, 99, 31, 13, 49, 99, 1, 186, 1, 152, 76, 17, 208, 425, 1, 19, 109, 699, 1, 584, 1, 433, 823, 23, 1, 1625, 437, 1140, 193, 746, 1, 2003, 1748, 2749, 244, 29, 1, 7404, 1, 31, 4158, 3258, 3766, 6307, 1
Offset: 1
From _Gus Wiseman_, Sep 24 2019: (Start)
The a(1) = 1 through a(12) = 15 strict integer partitions whose average is an integer (A = 10, B = 11, C = 12):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)
(31) (42) (53) (432) (64) (75)
(51) (62) (531) (73) (84)
(321) (71) (621) (82) (93)
(91) (A2)
(B1)
(543)
(642)
(651)
(732)
(741)
(831)
(921)
(5421)
(6321)
(End)
The BI-numbers of these partitions are given by
A326669 (numbers whose binary indices have integer mean).
Strict partitions with integer geometric mean are
A326625.
Strict partitions whose maximum divides their sum are
A326850.
-
a:= proc(m) option remember; local b; b:=
proc(n, i, t) option remember; `if`(i*(i+1)/2Alois P. Heinz, Sep 25 2019
-
npdp[n_]:=Count[Select[IntegerPartitions[n],Length[#]==Length[ Union[ #]]&], ?(Divisible[n,Length[#]]&)]; Array[npdp,70] (* _Harvey P. Dale, Feb 12 2016 *)
a[m_] := a[m] = Module[{b}, b[n_, i_, t_] := b[n, i, t] = If[i(i+1)/2 < n, 0, If[n == 0, If[Mod[m, t] == 0, 1, 0], b[n, i - 1, t] + b[n - i, Min[n - i, i - 1], t + 1]]]; If[PrimeQ[m], 1, b[m, m, 0]]];
Array[a, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)
A326836
Heinz numbers of integer partitions whose maximum part divides their sum.
Original entry on oeis.org
2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 150, 151
Offset: 1
The sequence of terms together with their prime indices begins:
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
12: {1,1,2}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
30: {1,2,3}
31: {11}
32: {1,1,1,1,1}
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Divisible[Total[primeMS[#]],Max[primeMS[#]]]&]
A326837
Heinz numbers of integer partitions whose length and maximum both divide their sum.
Original entry on oeis.org
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197
Offset: 1
The sequence of terms together with their prime indices begins:
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
30: {1,2,3}
31: {11}
32: {1,1,1,1,1}
37: {12}
Cf.
A001222,
A047993,
A056239,
A061395,
A067538,
A112798,
A316413,
A326836,
A326843,
A326847,
A326848.
-
isA326837 := proc(n)
psigsu := A056239(n) ;
psigma := A061395(n) ;
psigle := numtheory[bigomega](n) ;
if modp(psigsu,psigma) = 0 and modp(psigsu,psigle) = 0 then
true;
else
false;
end if;
end proc:
n := 1:
for i from 2 to 3000 do
if isA326837(i) then
printf("%d %d\n",n,i);
n := n+1 ;
end if;
end do: # R. J. Mathar, Aug 09 2019
-
Select[Range[2,100],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Divisible[Total[y],Max[y]]&&Divisible[Total[y],Length[y]]]&]
A326842
Number of integer partitions of n whose parts all divide n and whose length also divides n.
Original entry on oeis.org
1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 21, 2, 5, 6, 9, 2, 22, 2, 21, 6, 5, 2, 134, 3, 5, 6, 23, 2, 157, 2, 27, 6, 5, 6, 478, 2, 5, 6, 208, 2, 224, 2, 31, 63, 5, 2, 1720, 3, 30, 6, 34, 2, 322, 6, 295, 6, 5, 2, 13899, 2, 5, 68, 126, 8, 429, 2, 42, 6, 358, 2, 19959, 2
Offset: 0
The a(1) = 1 through a(8) = 5 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (11111) (33) (1111111) (44)
(1111) (222) (2222)
(321) (4211)
(111111) (11111111)
The a(12) = 21 partitions:
(12)
(6,6)
(4,4,4)
(6,3,3)
(6,4,2)
(3,3,3,3)
(4,3,3,2)
(4,4,2,2)
(4,4,3,1)
(6,2,2,2)
(6,3,2,1)
(6,4,1,1)
(2,2,2,2,2,2)
(3,2,2,2,2,1)
(3,3,2,2,1,1)
(3,3,3,1,1,1)
(4,2,2,2,1,1)
(4,3,2,1,1,1)
(4,4,1,1,1,1)
(6,2,1,1,1,1)
(1,1,1,1,1,1,1,1,1,1,1,1)
Partitions using divisors are
A018818.
Partitions whose length divides their sum are
A067538.
-
Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],Divisible[n,Length[#]]&]],{n,1,30}]
A326849
Number of integer partitions of n whose length times maximum is a multiple of n.
Original entry on oeis.org
1, 1, 2, 2, 3, 2, 6, 2, 5, 5, 10, 2, 19, 2, 18, 26, 24, 2, 55, 2, 87, 82, 60, 2, 207, 86, 106, 192, 363, 2, 668, 2, 527, 616, 304, 928, 1827, 2, 498, 1518, 3229, 2, 4294, 2, 4445, 6307, 1266, 2, 11560, 3629, 8280, 7802, 13633, 2, 19120, 18938, 31385, 16618, 4584
Offset: 0
The a(1) = 1 through a(9) = 5 partitions:
1 2 3 4 5 6 7 8 9
11 111 22 11111 33 1111111 44 333
1111 222 2222 621
411 4211 321111
3111 11111111 111111111
111111
For example, (4,1,1) is such a partition because its length times maximum is 3 * 4 = 12, which is a multiple of 6.
-
Table[If[n==0,1,Length[Select[IntegerPartitions[n],Divisible[Max[#]*Length[#],n]&]]],{n,0,30}]
A326851
Number of strict integer partitions of n whose length and maximum both divide n.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 5, 1, 6, 1, 1, 1, 16, 1, 1, 1, 12, 1, 33, 1, 15, 1, 1, 1, 60, 1, 1, 1, 51, 1, 81, 1, 31, 57, 1, 1, 216, 1, 55, 1, 45, 1, 230, 1, 223, 1, 1, 1, 800, 1, 1, 314, 273, 1, 607, 1, 81, 1, 315, 1, 2404, 1, 1, 319
Offset: 0
The a(6) = 2 through a(24) = 16 partitions (1 terms not shown):
6 12 15 16 18 20 24
3,2,1 6,4,2 5,4,3,2,1 8,4,3,1 9,5,4 10,5,3,2 12,7,5
6,5,1 8,5,2,1 9,6,3 10,5,4,1 12,8,4
6,3,2,1 9,7,2 10,6,3,1 12,9,3
9,8,1 10,7,2,1 12,10,2
10,4,3,2,1 12,11,1
8,7,5,4
8,7,6,3
12,5,4,3
12,6,4,2
12,6,5,1
12,7,3,2
12,7,4,1
12,8,3,1
12,9,2,1
8,6,4,3,2,1
-
Table[If[n==0,1,Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[n,Max[#]]&&Divisible[n,Length[#]]&]]],{n,0,30}]
A326848
Heinz numbers of integer partitions of m >= 0 whose length times maximum is a multiple of m.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 78, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181
Offset: 1
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
29: {10}
31: {11}
32: {1,1,1,1,1}
37: {12}
Cf.
A001222,
A047993,
A056239,
A061395,
A067538,
A112798,
A316413,
A326836,
A326843,
A326847,
A326849,
A326851.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],#==1||Divisible[Max[primeMS[#]]*Length[primeMS[#]],Total[primeMS[#]]]&]
A326850
Number of strict integer partitions of n whose maximum part divides n.
Original entry on oeis.org
0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 10, 1, 10, 5, 12, 1, 23, 1, 18, 15, 23, 1, 49, 1, 34, 36, 38, 1, 106, 1, 54, 79, 81, 1, 189, 1, 124, 162, 104, 1, 412, 1, 145, 307, 289, 1, 608, 12, 437, 559, 256, 1, 1432, 1, 340, 981, 976, 79, 1730, 1
Offset: 0
The initial terms count the following partitions:
1: (1)
2: (2)
3: (3)
4: (4)
5: (5)
6: (6)
6: (3,2,1)
7: (7)
8: (8)
8: (4,3,1)
9: (9)
10: (10)
10: (5,4,1)
10: (5,3,2)
11: (11)
12: (12)
12: (6,5,1)
12: (6,4,2)
12: (6,3,2,1)
13: (13)
14: (14)
14: (7,6,1)
14: (7,5,2)
14: (7,4,3)
14: (7,4,2,1)
15: (15)
15: (5,4,3,2,1)
Positions of 1's appear to be
A308168.
The non-strict case is given by
A067538.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[n,Max[#]]&]],{n,0,30}]
A340828
Number of strict integer partitions of n whose maximum part is a multiple of their length.
Original entry on oeis.org
1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777
Offset: 1
The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
1 2 3 4 5 6 7 8 9 A B C D E F G
21 41 42 43 62 63 64 65 84 85 86 87 A6
321 61 81 82 83 A2 A3 A4 A5 C4
621 631 A1 642 C1 C2 C3 E2
4321 632 651 643 653 E1 943
641 921 652 932 654 952
931 941 942 961
8321 951 C31
C21 8431
8421 8521
54321
Note: A-numbers of Heinz-number sequences are in parentheses below.
A072233 counts partitions by sum and length, with strict case
A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (
A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Max@@#,Length[#]]&]],{n,30}]
Showing 1-10 of 20 results.
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