A365314
Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 3, 6, 8, 14, 14, 23, 24, 33, 28, 52, 36, 55, 58, 73, 53, 95, 62, 110, 94, 105, 81, 165, 105, 133, 132, 176, 112, 225, 123, 210, 174, 192, 186, 306, 157, 223, 218, 328, 180, 354, 192, 324, 315, 288, 216, 474, 260, 383, 311, 404, 254, 491, 338, 511, 360
Offset: 0
We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
The a(2) = 1 through a(7) = 14 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(1,3) (1,3) (1,3) (1,3) (1,3)
(2,3) (1,4) (1,4) (1,4) (1,4)
(2,3) (1,5) (1,5) (1,5)
(2,4) (2,3) (1,6) (1,6)
(3,4) (2,5) (2,3) (1,7)
(3,5) (2,4) (2,3)
(4,5) (2,5) (2,5)
(2,6) (2,7)
(3,4) (3,4)
(3,5) (3,7)
(3,6) (4,7)
(4,6) (5,7)
(5,6) (6,7)
For all subsets instead of just pairs we have
A365073, complement
A365380.
The case of positive coefficients is
A365315, for all subsets
A088314.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}], combs[n,#]!={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365314(n):
a = set()
for i in range(1,n+1):
if not n%i:
a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
else:
for j in count(0,i):
if j > n:
break
k = n-j
for d in divisors(k):
if d>=i:
break
a.add((d,i))
return len(a) # Chai Wah Wu, Sep 12 2023
A365378
Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 1, 1, 4, 2, 9, 5, 13, 10, 28, 7, 45, 25, 51, 32, 101, 31, 148, 50, 166, 106, 291, 47, 374, 176, 450, 179, 721, 121, 963, 285, 1080, 474, 1534, 200, 2140, 712, 2407, 599, 3539, 481, 4546, 1014, 4885
Offset: 0
The partition (5,2,2) has distinct parts {2,5} and has 11 = 3*2 + 1*5, so is not counted under a(11).
The partition (4,2,2) cannot be linearly combined to obtain 9, so is counted under a(9).
The partition (4,2,2) has distinct parts {2,4} and has 10 = 5*2 + 0*4, so is not counted under a(10).
The a(3) = 1 through a(10) = 10 partitions:
(2) (3) (2) (4) (2) (3) (2) (3)
(3) (5) (3) (5) (4) (4)
(4) (4) (6) (5) (6)
(22) (5) (7) (6) (7)
(6) (33) (7) (8)
(22) (8) (9)
(33) (22) (33)
(42) (42) (44)
(222) (44) (63)
(62) (333)
(222)
(422)
(2222)
For positive coefficients we have
A365323.
The complement is counted by
A365379.
The relatively prime case is
A365382.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combs[n,Union[#]]=={}&]],{n,0,10}]
-
from sympy.utilities.iterables import partitions
def A365378(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n) for b in partitions(m) if not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
A365321
Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 2, 4, 6, 10, 13, 18, 24, 30, 37, 46, 54, 63, 77, 85, 99, 111, 127, 141, 161, 171, 194, 210, 235, 246, 277, 293, 322, 342, 372, 389, 428, 441, 491, 504, 545, 561, 612, 635, 680, 701, 753, 773, 836, 846, 911, 932, 1000, 1017, 1082, 1103, 1176, 1193
Offset: 0
For the pair p = (2,3) we have 4 = 2*2 + 0*3, so p is not counted under A365320(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is counted under a(4).
The a(2) = 1 through a(7) = 13 pairs:
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7)
(2,3) (2,3) (2,4) (2,3) (2,4)
(2,4) (2,5) (2,5) (2,6)
(3,4) (3,4) (2,6) (2,7)
(3,5) (3,4) (3,5)
(4,5) (3,5) (3,6)
(3,6) (3,7)
(4,5) (4,5)
(4,6) (4,6)
(5,6) (4,7)
(5,6)
(5,7)
(6,7)
For all subsets instead of just pairs we have
A365322, complement
A088314.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
Cf.
A070880,
A088571,
A088809,
A151897,
A326020,
A365043,
A365073,
A365311,
A365312,
A365378,
A365380.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}], combp[n,#]=={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365321(n):
a = set()
for i in range(1,n+1):
for j in count(i,i):
if j >= n:
break
for d in divisors(n-j):
if d>=i:
break
a.add((d,i))
return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 12 2023
A365379
Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 1, 3, 5, 10, 14, 27, 35, 61, 83, 128, 166, 264, 327, 482, 632, 882, 1110, 1565, 1938, 2663, 3339, 4401, 5471, 7290, 8921, 11555, 14291, 18280, 22303, 28507, 34507, 43534, 52882, 65798, 79621, 98932, 118629, 146072, 175562, 214708, 256351, 312583, 371779
Offset: 0
The partition (4,2,2) cannot be linearly combined to obtain 9, so is not counted under a(9). On the other hand, the same partition (4,2,2) has distinct parts {2,4} and has 10 = 1*2 + 2*4, so is counted under a(10).
The a(1) = 1 through a(5) = 14 partitions:
(1) (1) (1) (1) (1)
(2) (3) (2) (5)
(11) (11) (4) (11)
(21) (11) (21)
(111) (21) (31)
(22) (32)
(31) (41)
(111) (111)
(211) (211)
(1111) (221)
(311)
(1111)
(2111)
(11111)
For subsets with positive coefficients we have
A088314, complement
A088528.
The case of strict partitions with positive coefficients is also
A088314.
The complement is counted by
A365378.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n],combs[n,Union[#]]!={}&]],{n,0,10}]
-
from sympy.utilities.iterables import partitions
def A365379(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n+1) for b in partitions(m) if any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
A365315
Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 1, 2, 4, 5, 8, 10, 12, 15, 18, 20, 24, 28, 28, 35, 37, 42, 44, 49, 49, 60, 59, 66, 65, 79, 74, 85, 84, 93, 93, 107, 100, 120, 104, 126, 121, 142, 129, 145, 140, 160, 150, 173, 154, 189, 170, 196, 176, 208, 193, 223, 202, 238, 203, 241, 227, 267, 235
Offset: 0
We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
For the pair p = (2,3), we have 4 = 2*2 + 0*3, so p is counted under A365314(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is not counted under a(4).
The a(3) = 1 through a(10) = 15 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(1,3) (1,3) (1,3) (1,3) (1,3) (1,3) (1,3)
(1,4) (1,4) (1,4) (1,4) (1,4) (1,4)
(2,3) (1,5) (1,5) (1,5) (1,5) (1,5)
(2,4) (1,6) (1,6) (1,6) (1,6)
(2,3) (1,7) (1,7) (1,7)
(2,5) (2,3) (1,8) (1,8)
(3,4) (2,4) (2,3) (1,9)
(2,6) (2,5) (2,3)
(3,5) (2,7) (2,4)
(3,6) (2,6)
(4,5) (2,8)
(3,4)
(3,7)
(4,6)
For all subsets instead of just pairs we have
A088314, complement
A365322.
The case of nonnegative coefficients is
A365314, for all subsets
A365073.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
Cf.
A070880,
A088809,
A326020,
A364534,
A365043,
A365311,
A365312,
A365378,
A365379,
A365380,
A365383.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n],{2}],combp[n,#]!={}&]],{n,0,30}]
-
from itertools import count
from sympy import divisors
def A365315(n):
a = set()
for i in range(1,n+1):
for j in count(i,i):
if j >= n:
break
for d in divisors(n-j):
if d>=i:
break
a.add((d,i))
return len(a) # Chai Wah Wu, Sep 13 2023
A365382
Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 4, 2, 4, 12, 8, 20, 11, 14, 26, 43, 19, 38, 53, 51, 48, 101, 48, 124, 96, 121, 159, 134, 103, 241, 261, 244, 175, 401, 229, 488, 358, 328
Offset: 0
The a(11) = 2 through a(18) = 8 partitions:
(5,4) . (6,5) (6,5) (7,6) (7,5) (7,4) (7,5)
(7,3) (7,4) (8,5) (9,4) (7,6) (7,6) (8,7)
(7,5) (9,4) (9,5) (8,5) (10,7)
(8,3) (10,3) (11,3) (8,7) (11,4)
(9,5) (11,5)
(9,7) (12,5)
(10,3) (13,4)
(11,4) (7,5,5)
(11,5)
(13,3)
(7,4,4)
(10,3,3)
This is the relatively prime case of
A365378.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
combsu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],GCD@@#==1&&combsu[n,#]=={}&]],{n,0,20}]
-
from math import gcd
from sympy.utilities.iterables import partitions
def A365382(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n) for b in partitions(m) if gcd(*b.keys()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
A365323
Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.
Original entry on oeis.org
0, 0, 1, 1, 4, 3, 9, 7, 15, 16, 29, 23, 47, 43, 74, 65, 114, 100, 174, 153, 257, 228, 368, 312, 530, 454, 736, 645, 1025, 902, 1402, 1184, 1909, 1626, 2618, 2184, 3412, 2895, 4551, 3887, 5966, 5055, 7796, 6509, 10244, 8462, 13060, 10881, 16834, 14021, 21471
Offset: 1
The partition y = (3,3,2) has distinct parts {2,3}, and we have 9 = 3*2 + 1*3, so y is not counted under a(9).
The a(3) = 1 through a(10) = 16 partitions:
(2) (3) (2) (4) (2) (3) (2) (3)
(3) (5) (3) (5) (4) (4)
(4) (3,2) (4) (6) (5) (6)
(2,2) (5) (7) (6) (7)
(6) (3,3) (7) (8)
(2,2) (4,3) (8) (9)
(3,3) (5,2) (2,2) (3,3)
(4,2) (4,2) (4,4)
(2,2,2) (4,3) (5,2)
(4,4) (5,3)
(5,3) (5,4)
(6,2) (6,3)
(2,2,2) (7,2)
(4,2,2) (3,3,3)
(2,2,2,2) (4,3,2)
(5,2,2)
For strict partitions we have
A088528, nonnegative coefficients
A365312.
For length-2 subsets we have
A365321 (we use n instead of n-1).
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
-
combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combp[n,Union[#]]=={}&]],{n,10}]
-
from sympy.utilities.iterables import partitions
def A365323(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for k in range(1,n) for d in partitions(k) if tuple(sorted(set(d))) not in a) # Chai Wah Wu, Sep 12 2023
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