A325099 -id:A325099 - cp's OEIS frontend

A050315 Main diagonal of A050314.

1, 1, 1, 2, 1, 2, 2, 5, 1, 2, 2, 5, 2, 5, 5, 15, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15, 15, 52, 5, 15, 15, 52, 15, 52, 52, 203, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15

Offset: 0

Christian G. Bower, Sep 15 1999

nonn

Also, a(n) is the number of odd multinomial coefficients n!/(k_1!...k_m!) with 1 <= k_1 <= ... <= k_m and k_1 + ... + k_m = n. - Pontus von Brömssen, Mar 23 2018
From Gus Wiseman, Mar 30 2019: (Start)
Also the number of strict integer partitions of n with no binary carries. The Heinz numbers of these partitions are given by A325100. A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the a(1) = 1 through a(15) = 15 strict integer partitions with no binary carries are:
(1) (2) (3)  (4) (5)  (6)  (7)   (8) (9)  (A)  (B)   (C)  (D)   (E)   (F)
        (21)     (41) (42) (43)      (81) (82) (83)  (84) (85)  (86)  (87)
                           (52)                (92)       (94)  (A4)  (96)
                           (61)                (A1)       (C1)  (C2)  (A5)
                           (421)               (821)      (841) (842) (B4)
                                                                      (C3)
                                                                      (D2)
                                                                      (E1)
                                                                      (843)
                                                                      (852)
                                                                      (861)
                                                                      (942)
                                                                      (A41)
                                                                      (C21)
                                                                      (8421)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000110, A000120, A050314.
Cf. A070939, A080572, A247935, A267610.
Cf. A325093, A325095, A325096, A325099, A325100, A325103, A325110, A325123.
Main diagonal of A307431 and of A307505.

a:= n-> combinat[bell](add(i,i=convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 08 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],UnsameQ@@#&&stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)
a[n_] := BellB[DigitCount[n, 2, 1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021 *)

Bell number of number of 1's in binary: a(n) = A000110(A000120(n)).

A325096 Number of maximal subsets of {1...n} with no binary carries.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 12, 13, 14, 15, 15, 20, 25, 27, 32, 34, 36, 37, 42, 44, 46, 47, 49, 50, 51, 52, 52, 67, 82, 87, 102, 107, 112, 114, 129, 134, 139, 141, 146, 148, 150, 151, 166, 171, 176, 178, 183, 185, 187, 188, 193, 195, 197, 198, 200, 201

Offset: 0

Gus Wiseman, Mar 27 2019

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

			The a(1) = 1 through a(9) = 7 maximal subsets:
  {1}  {12}  {3}   {34}   {25}   {16}   {7}    {78}    {69}
             {12}  {124}  {34}   {25}   {16}   {168}   {78}
                          {124}  {34}   {25}   {258}   {168}
                                 {124}  {34}   {348}   {249}
                                        {124}  {1248}  {258}
                                                       {348}
                                                       {1248}

Alois P. Heinz, Table of n, a(n) for n = 0..8192

Cf. A000110, A019565, A050315, A080572, A247935, A267610, A267700.

Cf. A325095, A325099, A325100, A325101, A325103, A325104, A325107.

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}];
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]]],{n,0,10}]

a(2^n - 1) = A000110(n).

a(15)-a(61) from Alois P. Heinz, Mar 28 2019

A325118 Heinz numbers of binary carry-connected integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 19, 20, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 55, 59, 60, 61, 62, 64, 65, 67, 68, 71, 73, 75, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 92, 93, 94, 97, 100

Offset: 1

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose prime indices are binary carry-connected. 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.

			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}
  10: {1,3}
  11: {5}
  13: {6}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}

Cf. A019565, A048143, A056239, A112798, A247935, A304716, A305078.

Cf. A325098, A325099, A325105, A325109, A325110, A325119.

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]]]]]]]]];
Select[Range[100],Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]

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

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A050315, A080572, A247935, A267610, A267700.

Cf. A325096, A325099, A325104, A325106, A325108, A325110, A325118, A325119.

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 *)

a(21)-a(48) from Alois P. Heinz, Mar 29 2019

A325105 Number of binary carry-connected subsets of {1...n}.

Original entry on oeis.org

1, 2, 3, 7, 8, 20, 48, 112, 113, 325, 777, 1737, 3709, 7741, 15869, 32253, 32254, 96538, 225798, 485702, 1006338, 2049602, 4137346, 8315266, 16697102, 33465934, 67007886, 134100366, 268301518, 536720590, 1073575118, 2147316942, 2147316943, 6441886323

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. A subset is binary carry-connected if the graph whose vertices are the elements and whose edges are binary carries is connected.

			The a(0) = 1 through a(4) = 8 subsets:
  {}  {}   {}   {}       {}
      {1}  {1}  {1}      {1}
           {2}  {2}      {2}
                {3}      {3}
                {1,3}    {4}
                {2,3}    {1,3}
                {1,2,3}  {2,3}
                         {1,2,3}

Alois P. Heinz, Table of n, a(n) for n = 0..1023

Cf. A019565, A080572, A247935, A304714, A304716, A305078.
Cf. A325095, A325098, A325099, A325104, A325107, A325118, A325119.
Partial sums of A306299.

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, s) option remember; `if`(n=0,
      `if`(nops(s)>1, 0, 1), b(n-1, s)+b(n-1, g(n, s)))
    end:
a:= n-> b(n, {}):
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 31 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[Subsets[Range[n]],Length[csm[binpos/@#]]<=1&]],{n,0,10}]

a(n) = A306297(n,0) + A306297(n,1). - Alois P. Heinz, Mar 31 2019

a(16)-a(33) from Alois P. Heinz, Mar 31 2019

A325119 Heinz numbers of binary carry-connected strict integer partitions.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 15, 17, 19, 22, 23, 29, 30, 31, 34, 37, 39, 41, 43, 46, 47, 51, 53, 55, 59, 61, 62, 65, 67, 71, 73, 77, 79, 82, 83, 85, 87, 89, 91, 93, 94, 97, 101, 102, 103, 107, 109, 110, 113, 115, 118, 119, 127, 129, 130, 131, 134, 137, 139, 141

Offset: 1

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. An integer partition is binary carry-connected if the graph whose vertices are the parts and whose edges are binary carries is connected.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices are binary carry-connected. 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.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
  10: {1,3}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  22: {1,5}
  23: {9}
  29: {10}
  30: {1,2,3}
  31: {11}
  34: {1,7}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}

Cf. A019565, A048143, A056239, A112798, A304714, A304716, A305078.

Cf. A325098, A325099, A325100, A325105, A325110, A325118.

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]]]]]]]]];
Select[Range[100],SquareFreeQ[#]&&Length[csm[binpos/@PrimePi/@First/@FactorInteger[#]]]<=1&]

cp's OEIS Frontend

A050315 Main diagonal of A050314.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A325096 Number of maximal subsets of {1...n} with no binary carries.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A325118 Heinz numbers of binary carry-connected integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325098 Number of binary carry-connected integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A325105 Number of binary carry-connected subsets of {1...n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A325119 Heinz numbers of binary carry-connected strict integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica