A325106
Number of divisible binary-containment pairs of positive integers up to n.
Original entry on oeis.org
0, 0, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 31, 32, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 56, 57, 58, 61, 63, 64, 65, 66, 67, 70, 71, 72, 77, 77, 78, 79, 80, 81
Offset: 0
The a(3) = 1 through a(12) = 8 pairs:
{1,3} {1,3} {1,3} {1,3} {1,3} {1,3} {1,3} {1,3} {1,3} {1,3}
{1,5} {1,5} {1,5} {1,5} {1,5} {1,5} {1,5} {1,5}
{2,6} {1,7} {1,7} {1,7} {1,7} {1,7} {1,7}
{2,6} {2,6} {1,9} {1,9} {1,9} {1,9}
{2,6} {2,6} {2,6} {2,6}
{2,10} {1,11} {1,11}
{2,10} {2,10}
{4,12}
-
Table[Length[Select[Subsets[Range[n],{2}],Divisible[#[[2]],#[[1]]]&&SubsetQ[Position[Reverse[IntegerDigits[#[[2]],2]],1],Position[Reverse[IntegerDigits[#1[[1]],2]],1]]&]],{n,0,30}]
A325103
Number of increasing pairs of positive integers up to n with no binary carries.
Original entry on oeis.org
0, 0, 1, 1, 4, 5, 6, 6, 13, 16, 19, 20, 23, 24, 25, 25, 40, 47, 54, 57, 64, 67, 70, 71, 78, 81, 84, 85, 88, 89, 90, 90, 121, 136, 151, 158, 173, 180, 187, 190, 205, 212, 219, 222, 229, 232, 235, 236, 251, 258, 265, 268, 275, 278, 281, 282, 289, 292, 295, 296
Offset: 0
The a(2) = 1 through a(9) = 16 pairs:
{1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2} {1,2}
{1,4} {1,4} {1,4} {1,4} {1,4} {1,4}
{2,4} {2,4} {1,6} {1,6} {1,6} {1,6}
{3,4} {2,5} {2,4} {2,4} {1,8} {1,8}
{3,4} {2,5} {2,5} {2,4} {2,4}
{3,4} {3,4} {2,5} {2,5}
{2,8} {2,8}
{3,4} {2,9}
{3,8} {3,4}
{4,8} {3,8}
{5,8} {4,8}
{6,8} {4,9}
{7,8} {5,8}
{6,8}
{6,9}
{7,8}
-
Table[Length[Select[Subsets[Range[n],{2}],Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,30}]
A325104
Number of increasing pairs of positive integers up to n with at least one binary carry.
Original entry on oeis.org
0, 0, 0, 2, 2, 5, 9, 15, 15, 20, 26, 35, 43, 54, 66, 80, 80, 89, 99, 114, 126, 143, 161, 182, 198, 219, 241, 266, 290, 317, 345, 375, 375, 392, 410, 437, 457, 486, 516, 551, 575, 608, 642, 681, 717, 758, 800, 845, 877, 918, 960, 1007, 1051, 1100, 1150, 1203
Offset: 0
The a(3) = 2 through a(8) = 15 pairs:
{1,3} {1,3} {1,3} {1,3} {1,3} {1,3}
{2,3} {2,3} {1,5} {1,5} {1,5} {1,5}
{2,3} {2,3} {1,7} {1,7}
{3,5} {2,6} {2,3} {2,3}
{4,5} {3,5} {2,6} {2,6}
{3,6} {2,7} {2,7}
{4,5} {3,5} {3,5}
{4,6} {3,6} {3,6}
{5,6} {3,7} {3,7}
{4,5} {4,5}
{4,6} {4,6}
{4,7} {4,7}
{5,6} {5,6}
{5,7} {5,7}
{6,7} {6,7}
-
Table[Length[Select[Subsets[Range[n],{2}],Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]!={}&]],{n,0,30}]
A325098
Number of binary carry-connected integer partitions of n.
Original entry on oeis.org
1, 1, 2, 2, 4, 4, 7, 7, 13, 15, 23, 27, 42, 50, 72, 88, 125, 153, 211, 258, 349, 430, 569, 698, 914, 1119, 1444, 1765, 2252, 2745, 3470, 4214, 5276, 6387, 7934, 9568, 11800, 14181, 17379, 20818, 25351, 30264, 36668, 43633, 52589, 62394, 74872, 88576, 105818
Offset: 0
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (322) (44)
(31) (311) (51) (331) (53)
(1111) (11111) (222) (511) (62)
(321) (3211) (71)
(3111) (31111) (332)
(111111) (1111111) (2222)
(3221)
(3311)
(5111)
(32111)
(311111)
(11111111)
-
h:= proc(n, s) local i, m; m:= n;
for i in s do m:= Bits[Or](m, i) od; {m}
end:
g:= (n, s)-> (w-> `if`(w={}, s union {n}, s minus w union
h(n, w)))(select(x-> Bits[And](n, x)>0, s)):
b:= proc(n, i, s) option remember; `if`(n=0, `if`(nops(s)>1, 0, 1),
`if`(i<1, 0, b(n, i-1, s)+ b(n-i, min(i, n-i), g(i, s))))
end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 29 2019
-
binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n],Length[csm[binpos/@#]]<=1&]],{n,0,20}]
(* Second program: *)
h[n_, s_] := Module[{i, m = n}, Do[m = BitOr[m, i], {i, s}]; {m}];
g[n_, s_] := Function[w, If[w == {}, s ~Union~ {n}, (s ~Complement~ w) ~Union~
h[n, w]]][Select[s, BitAnd[n, #] > 0&]];
b[n_, i_, s_] := b[n, i, s] = If[n == 0, If[Length[s] > 1, 0, 1],
If[i < 1, 0, b[n, i - 1, s] + b[n - i, Min[i, n - i], g[i, s]]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 50] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *)
A325110
Number of strict integer partitions of n with no binary containments.
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 2, 5, 2, 3, 2, 6, 3, 6, 7, 15, 8, 10, 6, 13, 6, 10, 12, 23, 13, 16, 16, 26, 21, 30, 37, 60, 43, 52, 42, 60, 42, 50, 54, 81, 59, 60, 66, 80, 74, 86, 108, 145, 119, 125, 126, 144, 134, 140, 170, 208, 189, 193, 221, 248, 253, 292, 323, 435, 392
Offset: 0
The a(1) = 1 through a(12) = 3 partitions (A = 10, B = 11, C = 12):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)
(21) (41) (42) (43) (53) (63) (82) (65) (84)
(52) (81) (83) (93)
(61) (92)
(421) (A1)
(821)
-
binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]],{n,0,30}]
A325109
Number of integer partitions of n whose distinct parts have no binary containments.
Original entry on oeis.org
1, 1, 2, 3, 4, 5, 8, 10, 12, 15, 18, 23, 28, 32, 41, 52, 57, 66, 76, 90, 99, 117, 131, 157, 172, 194, 216, 255, 276, 313, 358, 410, 447, 511, 546, 630, 677, 750, 818, 945, 990, 1108, 1200, 1338, 1429, 1606, 1713, 1928, 2062, 2263, 2412, 2725, 2847, 3142, 3389
Offset: 0
The a(1) = 1 through a(8) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (41) (33) (43) (44)
(111) (211) (221) (42) (52) (53)
(1111) (2111) (222) (61) (422)
(11111) (411) (421) (611)
(2211) (2221) (2222)
(21111) (4111) (4211)
(111111) (22111) (22211)
(211111) (41111)
(1111111) (221111)
(2111111)
(11111111)
-
c:= proc() option remember; local i, x, y;
x, y:= map(n-> Bits[Split](n), [args])[];
for i to nops(x) do
if x[i]=1 and y[i]=0 then return false fi
od; true
end:
b:= proc(n, i, s) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, s)+`if`(ormap(j-> c(i, j), s), 0, add(
b(n-i*j, i-1, s union {i}), j=1..n/i))))
end:
a:= n-> b(n$2, {}):
seq(a(n), n=0..55); # Alois P. Heinz, Mar 29 2019
-
binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]],{n,0,15}]
(* Second program: *)
c[args_List] := c[args] = Module[{i, x, y}, {x, y} = Reverse@IntegerDigits[#, 2]& /@ args; For[i = 1, i <= Length[x], i++, If[x[[i]] == 1 && y[[i]] == 0, Return[False]]]; True];
b[n_, i_, s_List] := b[n, i, s] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, s] + If[AnyTrue[s, c[{i, #}]&], 0, Sum[b[n - i*j, i-1, s ~Union~ {i}], {j, 1, n/i}]]]];
a[n_] := b[n, n, {}];
a /@ Range[0, 55] (* Jean-François Alcover, Jun 03 2021, after Alois P. Heinz *)
A325101
Number of divisible binary-containment pairs of positive integers up to n.
Original entry on oeis.org
0, 1, 2, 4, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 24, 28, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 53, 55, 57, 61, 63, 64, 66, 68, 70, 72, 74, 76, 79, 81, 83, 85, 87, 89, 93, 95, 97, 99, 101, 103, 107, 109, 111, 115, 118, 120, 122, 124, 126, 130, 132, 134
Offset: 0
The a(1) = 1 through a(8) = 12 pairs:
(1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1) (1,1)
(2,2) (1,3) (1,3) (1,3) (1,3) (1,3) (1,3)
(2,2) (2,2) (1,5) (1,5) (1,5) (1,5)
(3,3) (3,3) (2,2) (2,2) (1,7) (1,7)
(4,4) (3,3) (2,6) (2,2) (2,2)
(4,4) (3,3) (2,6) (2,6)
(5,5) (4,4) (3,3) (3,3)
(5,5) (4,4) (4,4)
(6,6) (5,5) (5,5)
(6,6) (6,6)
(7,7) (7,7)
(8,8)
-
Table[Length[Select[Tuples[Range[n],2],Divisible[#[[2]],#[[1]]]&&SubsetQ[Position[Reverse[IntegerDigits[#[[2]],2]],1],Position[Reverse[IntegerDigits[#1[[1]],2]],1]]&]],{n,0,30}]
A325107
Number of subsets of {1...n} with no binary containments.
Original entry on oeis.org
1, 2, 4, 5, 10, 13, 18, 19, 38, 52, 77, 83, 133, 147, 166, 167, 334, 482, 764, 848, 1465, 1680, 1987, 2007, 3699, 4413, 5488, 5572, 7264, 7412, 7579, 7580, 15160, 22573, 37251, 42824, 77387, 92863, 116453, 118461, 227502, 286775, 382573, 392246, 555661, 574113
Offset: 0
The a(0) = 1 through a(6) = 18 subsets:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{1,2} {3} {3} {3} {3}
{1,2} {4} {4} {4}
{1,2} {5} {5}
{1,4} {1,2} {6}
{2,4} {1,4} {1,2}
{3,4} {2,4} {1,4}
{1,2,4} {2,5} {1,6}
{3,4} {2,4}
{3,5} {2,5}
{1,2,4} {3,4}
{3,5}
{3,6}
{5,6}
{1,2,4}
{3,5,6}
-
c:= proc() option remember; local i, x, y;
x, y:= map(n-> Bits[Split](n), [args])[];
for i to nops(x) do
if x[i]=1 and y[i]=0 then return false fi
od; true
end:
b:= proc(n, s) option remember; `if`(n=0, 1, b(n-1, s)+
`if`(ormap(i-> c(n, i), s), 0, b(n-1, s union {n})))
end:
a:= n-> b(n, {}):
seq(a(n), n=0..34); # Alois P. Heinz, Mar 28 2019
-
binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Range[n]],stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]],{n,0,13}]
A325102
Number of ordered pairs of positive integers up to n with no binary carries.
Original entry on oeis.org
0, 0, 2, 2, 8, 10, 12, 12, 26, 32, 38, 40, 46, 48, 50, 50, 80, 94, 108, 114, 128, 134, 140, 142, 156, 162, 168, 170, 176, 178, 180, 180, 242, 272, 302, 316, 346, 360, 374, 380, 410, 424, 438, 444, 458, 464, 470, 472, 502, 516, 530, 536, 550, 556, 562, 564, 578
Offset: 0
The a(2) = 2 through a(6) = 12 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(2,1) (2,1) (1,4) (1,4) (1,4) (1,4)
(2,1) (2,1) (1,6) (1,6)
(2,4) (2,4) (2,1) (2,1)
(3,4) (2,5) (2,4) (2,4)
(4,1) (3,4) (2,5) (2,5)
(4,2) (4,1) (3,4) (3,4)
(4,3) (4,2) (4,1) (4,1)
(4,3) (4,2) (4,2)
(5,2) (4,3) (4,3)
(5,2) (5,2)
(6,1) (6,1)
-
Table[Length[Select[Tuples[Range[n],2],Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,30}]
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