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Search: a237258 -id:a237258
Displaying 1-10 of 39 results found. page 1 2 3 4
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A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.
(Formerly M2574 N1018) 		+10
62
1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018.
FORMULA
See A213074 for Metropolis and Stein's formulas.
a(n) = A000041(2*n) - A006827(n) = A000041(2*n) - A046663(2*n,n).
a(n) = A276107(2*n). - Max Alekseyev, Oct 17 2022
EXAMPLE
Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). - N. J. A. Sloane, Jun 03 2012
From Gus Wiseman, Oct 27 2022: (Start)
The a(1) = 1 through a(4) = 14 partitions:
(11) (22) (33) (44)
(211) (321) (422)
(1111) (2211) (431)
(3111) (2222)
(21111) (3221)
(111111) (3311)
(4211)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
(End)
MAPLE
g:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
end:
b:= proc(n, i, s) option remember;
`if`(i=1 and s<>{} or n in s, g(n, i), `if`(i<1 or s={}, 0,
b(n, i-1, s)+ `if`(i>n, 0, b(n-i, i, map(x-> {`if`(x>n-i, NULL,
max(x, n-i-x)), `if`(x<i or x>n, NULL, max(x-i, n-x))}[], s)))))
end:
a:= n-> b(2*n, n, {n}):
seq(a(n), n=1..25); # Alois P. Heinz, Jul 10 2012
MATHEMATICA
b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *)
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
subptns[s_]:=primeMS/@Divisors[Times@@Prime/@s];
Table[Length[Select[IntegerPartitions[2n], MemberQ[Total/@subptns[#], n]&]], {n, 10}] (* Gus Wiseman, Oct 27 2022 *)
PROG
(Python)
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
def A002219(n): return len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)), 2)}) # Chai Wah Wu, Sep 20 2023
CROSSREFS
Column m=2 of A213086.
Bisection of A276107.
Cf. A064914, A000041, A002220, A002221, A002222, A213074, A006827, A046663.
The strict version is A237258, ranked by A357854.
Ranked by A357976 = positions of nonzero terms in A357879.
A122768 counts distinct submultisets of partitions.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A108917, A235130, A237194, A300061.
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
Better description from Vladeta Jovovic, Mar 06 2000
More terms from Christian G. Bower, Oct 12 2001
Edited by N. J. A. Sloane, Jun 03 2012
More terms from Alois P. Heinz, Jul 10 2012
STATUS
approved
A006827 Number of partitions of 2n with all subsums different from n.
(Formerly M1351) 		+10
54
1, 2, 5, 8, 17, 24, 46, 64, 107, 147, 242, 302, 488, 629, 922, 1172, 1745, 2108, 3104, 3737, 5232, 6419, 8988, 10390, 14552, 17292, 23160, 27206, 36975, 41945, 57058, 65291, 85895, 99384, 130443, 145283, 193554, 218947, 281860, 316326, 413322, 454229, 594048 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Partitions of this type are also called non-biquanimous partitions. - Gus Wiseman, Apr 19 2024
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.
FORMULA
a(n) = A000041(2*n) - A002219(n).
a(n) = A046663(2*n,n).
EXAMPLE
From Gus Wiseman, Apr 19 2024: (Start)
The a(1) = 1 through a(5) = 17 partitions (A = 10):
(2) (4) (6) (8) (A)
(31) (42) (53) (64)
(51) (62) (73)
(222) (71) (82)
(411) (332) (91)
(521) (433)
(611) (442)
(5111) (622)
(631)
(721)
(811)
(3331)
(4222)
(6211)
(7111)
(22222)
(61111)
(End)
MAPLE
b:= proc(n, i, s) option remember;
`if`(0 in s or n in s, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, s)+
`if`(i<=n, b(n-i, i, select(y-> 0<=y and y<=n-i,
map(x-> [x, x-i][], s))), 0))))
end:
a:= n-> b(2*n, 2*n, {n}):
seq(a(n), n=1..25); # Alois P. Heinz, Jul 10 2012
MATHEMATICA
b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; a[n_] := b[2*n, 2*n, {n}]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *)
PROG
(Python)
from itertools import combinations_with_replacement
from collections import Counter
from sympy import npartitions
from sympy.utilities.iterables import partitions
def A006827(n): return npartitions(n<<1)-len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)), 2)}) # Chai Wah Wu, Sep 20 2023
CROSSREFS
The complement is counted by A002219, ranks A357976.
Central diagonal of A046663.
The strict case is A321142, even bisection of A371794 (odd A078408).
This is the "bi-" version of A321451, ranks A321453.
Column k = 0 of A367094.
These partitions have Heinz numbers A371731.
Even bisection of A371795 (odd A058695).
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A237258, A305551, A321452, A365543, A365663, A366320, A371736, A371782, A371792.
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Don Reble, Nov 03 2001
More terms from Alois P. Heinz, Jul 10 2012
STATUS
approved
A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k. +10
46
1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.
Rows are palindromic.
Are there only two zeros in the whole triangle?
LINKS
Table of n, a(n) for n=0..77.
EXAMPLE
Triangle begins:
1
1 1
1 0 1
2 1 1 2
2 1 0 1 2
3 1 1 1 1 3
4 2 2 1 2 2 4
5 2 2 2 2 2 2 5
6 3 2 3 1 3 2 3 6
8 3 3 4 3 3 4 3 3 8
Row n = 6 counts the following strict partitions:
(6) (5,1) (4,2) (3,2,1) (4,2) (5,1) (6)
(5,1) (3,2,1) (3,2,1) (3,2,1) (3,2,1) (5,1)
(4,2) (4,2)
(3,2,1) (3,2,1)
Row n = 10 counts the following strict partitions:
A 91 82 73 64 532 64 73 82 91 A
64 541 532 532 541 541 541 532 532 541 64
73 631 721 631 631 4321 631 631 721 631 73
82 721 4321 721 4321 4321 721 4321 721 82
91 4321 4321 4321 4321 91
532 532
541 541
631 631
721 721
4321 4321
MATHEMATICA
Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#], k]&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
Columns k = 0 and k = n are A000009.
The non-strict complement is A046663, central column A006827.
Central column n = 2k is A237258.
For subsets instead of partitions we have A365381.
The non-strict case is A365543.
The complement is A365663.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 16 2023
STATUS
approved
A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k. +10
45
1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
2,5
COMMENTS
Warning: Do not confuse with the non-strict version A046663.
Rows are palindromes.
LINKS
Table of n, a(n) for n=2..92.
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.
EXAMPLE
Triangle begins:
1
1 1
1 2 1
2 2 2 2
2 2 3 2 2
3 3 3 3 3 3
3 4 3 5 3 4 3
5 5 4 5 5 4 5 5
5 6 5 6 7 6 5 6 5
7 7 7 7 7 7 7 7 7 7
8 9 8 8 8 11 8 8 8 9 8
Row n = 8 counts the following strict partitions:
(8) (8) (8) (8) (8) (8) (8)
(6,2) (7,1) (7,1) (7,1) (7,1) (7,1) (6,2)
(5,3) (5,3) (6,2) (6,2) (6,2) (5,3) (5,3)
(4,3,1) (5,3) (4,3,1)
(5,2,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#], k]&]], {n, 2, 15}, {k, 1, n-1}]
CROSSREFS
Columns k = 0 and k = n are A025147.
The non-strict version is A046663, central column A006827.
Central column n = 2k is A321142.
The complement for subsets instead of strict partitions is A365381.
The complement is A365661, non-strict A365543, central column A237258.
Row sums are A365922.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364272 counts sum-full strict partitions, sum-free A364349.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 17 2023
STATUS
approved
A357976 Numbers with a divisor having the same sum of prime indices as their quotient. +10
38
1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
The terms together with their prime indices begin:
1: {}
4: {1,1}
9: {2,2}
12: {1,1,2}
16: {1,1,1,1}
25: {3,3}
30: {1,2,3}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
49: {4,4}
For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.
MAPLE
filter:= proc(n) local F, s, t, i, R;
F:= ifactors(n)[2];
F:= map(t -> [numtheory:-pi(t[1]), t[2]], F);
s:= add(t[1]*t[2], t=F)/2;
if not s::integer then return false fi;
try
R:= Optimization:-Maximize(0, [add(F[i][1]*x[i], i=1..nops(F)) = s, seq(x[i]<= F[i][2], i=1..nops(F))], assume=nonnegint, depthlimit=20);
catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
end try;
true
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Oct 26 2023
MATHEMATICA
sumprix[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimePi[p]]];
Select[Range[100], MemberQ[sumprix/@Divisors[#], sumprix[#]/2]&]
CROSSREFS
The partitions with these Heinz numbers are counted by A002219.
A subset of A300061.
The squarefree case is A357854, counted by A237258.
Positions of nonzero terms in A357879.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 26 2022
STATUS
approved
A064914 Number of ordered biquanimous partitions of 2n. +10
33
1, 1, 5, 23, 105, 449, 1902, 7828, 31976, 129200, 520425, 2088217, 8371186, 33514797, 134140430, 536699674, 2147154667, 8589198795, 34358341823, 137435830265, 549749857574, 2199010044813, 8796067657649, 35184315676573, 140737380485376, 562949713881526 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
A biquanimous partition is one that can be bisected into two equal sized parts: e.g. 3+2+1 is a biquanimous partition of 6 as it contains 3 and 2+1, but 5+1 is not.
LINKS
Table of n, a(n) for n=0..25.
EXAMPLE
From Gus Wiseman, Apr 19 2024: (Start)
The a(0) = 1 through a(3) = 23 biquanimous compositions:
() (11) (22) (33)
(112) (123)
(121) (132)
(211) (213)
(1111) (231)
(312)
(321)
(1113)
(1122)
(1131)
(1212)
(1221)
(1311)
(2112)
(2121)
(2211)
(3111)
(11112)
(11121)
(11211)
(12111)
(21111)
(111111)
(End)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], MemberQ[Total/@Subsets[#], n]&]], {n, 0, 5}] (* Gus Wiseman, Apr 19 2024 *)
CROSSREFS
The unordered version (integer partitions) is A002219, ranks A357976.
The unordered complement is A371795, even case A006827, ranks A371731.
The complement is counted by A371956.
These compositions have ranks A372120, complement A372119.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.
KEYWORD
nonn
AUTHOR
Christian G. Bower, Oct 12 2001
EXTENSIONS
More terms from Alois P. Heinz, Jun 12 2017
STATUS
approved
A371783 Irregular triangle read by rows where T(n,d) is the number of integer partitions of n that can be partitioned into d blocks with equal sums, with d ranging over all divisors d|n. +10
31
1, 2, 1, 3, 1, 5, 3, 1, 7, 1, 11, 6, 4, 1, 15, 1, 22, 14, 5, 1, 30, 10, 1, 42, 25, 6, 1, 56, 1, 77, 53, 30, 15, 7, 1, 101, 1, 135, 89, 8, 1, 176, 65, 21, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
These could be called d-quanimous partitions, cf. A002219, A064914, A321452.
LINKS
Table of n, a(n) for n=1..45.
EXAMPLE
Triangle begins:
1
2 1
3 1
5 3 1
7 1
11 6 4 1
15 1
22 14 5 1
30 10 1
42 25 6 1
56 1
77 53 30 15 7 1
101 1
135 89 8 1
176 65 21 1
Row n = 6 counts the following partitions:
(6) (33) (222) (111111)
(33) (321) (2211)
(42) (2211) (21111)
(51) (3111) (111111)
(222) (21111)
(321) (111111)
(411)
(2211)
(3111)
(21111)
(111111)
MATHEMATICA
hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]], {n, 1, 8}, {k, Divisors[n]}]
CROSSREFS
Row lengths are A000005.
Column k = 1 is A000041.
Inserting zeros gives A371954.
Row sums are A372121.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371736 counts non-quanimous strict partitons, complement A371737.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A035470, A064914, A321455, A365543, A371791, A371795.
KEYWORD
nonn,tabf,more
AUTHOR
Gus Wiseman, Apr 14 2024
STATUS
approved
A321142 Number of strict integer partitions of 2*n with no subset summing to n. +10
30
0, 1, 2, 3, 5, 7, 11, 15, 23, 30, 43, 57, 79, 102, 138, 174, 232, 292, 375, 471, 602, 741, 935, 1148, 1425, 1733, 2137, 2571, 3156, 3789, 4557, 5470, 6582, 7796, 9317, 11027, 13058, 15400, 18159, 21249, 24971, 29170, 33986, 39596, 46073, 53219, 61711, 71330, 82171 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200
EXAMPLE
The a(1) = 1 through a(8) = 23 partitions:
(2) (4) (6) (8) (10) (12) (14) (16)
(3,1) (4,2) (5,3) (6,4) (7,5) (8,6) (9,7)
(5,1) (6,2) (7,3) (8,4) (9,5) (10,6)
(7,1) (8,2) (9,3) (10,4) (11,5)
(5,2,1) (9,1) (10,2) (11,3) (12,4)
(6,3,1) (11,1) (12,2) (13,3)
(7,2,1) (5,4,3) (13,1) (14,2)
(7,3,2) (6,5,3) (15,1)
(7,4,1) (8,4,2) (7,5,4)
(8,3,1) (8,5,1) (7,6,3)
(9,2,1) (9,3,2) (9,4,3)
(9,4,1) (9,5,2)
(10,3,1) (9,6,1)
(11,2,1) (10,4,2)
(8,3,2,1) (10,5,1)
(11,3,2)
(11,4,1)
(12,3,1)
(13,2,1)
(6,5,4,1)
(7,4,3,2)
(9,4,2,1)
(10,3,2,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], And[UnsameQ@@#, !Or@@Table[SameQ[Total[#[[s]]], n/2], {s, Subsets[Range[Length[#]]]}]]&]], {n, 2, 20, 2}]
CROSSREFS
Cf. A002219, A006827, A108917, A237258, A275972, A279791, A284640, A305551.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 28 2018
EXTENSIONS
a(33)-a(48) from Giovanni Resta, Oct 30 2018
STATUS
approved
A371795 Number of non-biquanimous integer partitions of n. +10
27
0, 1, 1, 3, 2, 7, 5, 15, 8, 30, 17, 56, 24, 101, 46, 176, 64, 297, 107, 490, 147, 792, 242, 1255, 302, 1958, 488, 3010, 629, 4565, 922 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.
LINKS
Table of n, a(n) for n=0..30.
EXAMPLE
The a(1) = 1 through a(8) = 8 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(111) (41) (51) (52) (62)
(221) (222) (61) (71)
(311) (411) (322) (332)
(2111) (331) (521)
(11111) (421) (611)
(511) (5111)
(2221)
(3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
MATHEMATICA
biqQ[y_]:=MemberQ[Total/@Subsets[y], Total[y]/2];
Table[Length[Select[IntegerPartitions[n], Not@*biqQ]], {n, 0, 15}]
CROSSREFS
The complement is counted by A002219 aerated, ranks A357976.
Even bisection is A006827, odd A058695.
The strict complement is A237258, ranks A357854.
This is the "bi-" version of A321451, ranks A321453.
The complement is the "bi-" version of A321452, ranks A321454.
These partitions have ranks A371731.
The strict case is A371794, bisections A321142, A078408.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371736 counts non-quanimous strict partitons, complement A371737.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A035470, A064914, A305551, A336137, A365543, A365661, A365663, A366320, A365925, A367094, A371788.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Apr 07 2024
STATUS
approved
A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient. +10
25
1, 30, 70, 154, 165, 210, 273, 286, 390, 442, 462, 561, 595, 646, 714, 741, 858, 874, 910, 1045, 1155, 1173, 1254, 1326, 1330, 1334, 1495, 1653, 1771, 1794, 1798, 1870, 1938, 2139, 2145, 2294, 2415, 2465, 2470, 2530, 2622, 2639, 2730, 2926, 2945, 2958, 3034 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
LINKS
Table of n, a(n) for n=1..47.
EXAMPLE
The terms together with their prime indices begin:
1: {}
30: {1,2,3}
70: {1,3,4}
154: {1,4,5}
165: {2,3,5}
210: {1,2,3,4}
273: {2,4,6}
286: {1,5,6}
390: {1,2,3,6}
For example, 210 has factorization 14*15, and both factors have the same sum of prime indices 5, so 210 is in the sequence.
MATHEMATICA
sumprix[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimePi[p]]];
Select[Range[1000], SquareFreeQ[#]&&MemberQ[sumprix/@Divisors[#], sumprix[#]/2]&]
CROSSREFS
The partitions with these Heinz numbers are counted by A237258.
A subset of A319241, squarefree case of A300061.
Squarefree positions of nonzero terms in A357879.
This is the squarefree case of A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A235130, A237194, A276107, A300273, A321144, A357975, A357976.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 27 2022
STATUS
approved
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