A325097 -id:A325097 - cp's OEIS frontend

A247935 Number of integer partitions of n whose distinct parts have no binary carries.

1, 1, 2, 3, 4, 5, 8, 10, 11, 14, 18, 21, 26, 30, 38, 49, 47, 55, 66, 74, 84, 96, 110, 126, 134, 151, 171, 195, 209, 235, 272, 318, 307, 349, 377, 422, 448, 491, 534, 595, 617, 674, 734, 801, 841, 925, 998, 1098, 1118, 1219, 1299, 1418, 1476, 1591, 1711, 1865

Offset: 0

David S. Newman, Sep 26 2014

nonn

From Gus Wiseman, Mar 31 2019: (Start)
A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the reversed binary expansions of 2, 5, and 8 are
  {0,1}
  {1,0,1}
  {0,0,0,1}
and since there are no columns with more than one 1, the partition (8,5,2) is counted under a(15). The Heinz numbers of these partitions are given by A325097.
(End)

			From _Gus Wiseman_, Mar 30 2019: (Start)
The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (33)      (43)       (44)
             (111)  (211)   (221)    (42)      (52)       (422)
                    (1111)  (2111)   (222)     (61)       (611)
                            (11111)  (411)     (421)      (2222)
                                     (2211)    (2221)     (4211)
                                     (21111)   (4111)     (22211)
                                     (111111)  (22111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000110, A000120, A050315, A070939, A080572, A248605, A267610.

Cf. A325095, A325096, A325097, A325098, A325102, A325103, A325109.

with(Bits):
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, t) +`if`(i>n or And(t, i)>0, 0,
      add(b(n-i*j, i-1, Or(t, i)), j=1..n/i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2014

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[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n || BitAnd[t, i] > 0, 0, Sum[b[n - i*j, i - 1, BitOr[t, i]], {j, 1, n/i}]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 80] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Oct 15 2014

Name edited by Gus Wiseman, Mar 31 2019

A371452 Number of connected components of the prime indices of the binary indices of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5

Offset: 1

Gus Wiseman, Apr 01 2024

nonn

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.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The prime indices of binary indices of 281492156579880 are {{1,1},{1,2},{3,4},{4,4}}, with 2 connected components {{1,1},{1,2}} and {{3,4},{4,4}}, so a(281492156579880) = 2.

Positions of first appearances are A080355, opposite A325782.
For prime indices of prime indices we have A305079, ones A305078.
For binary indices of binary indices we have A326753, ones A326749.
Positions of ones are A371291.
For binary indices of prime indices we have A371451, ones A325118.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A326964 counts connected set-systems, covering A323818.
Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[csm[prix/@bix[n]]],{n,100}]

A325095 Number of subsets of {1...n} with no binary carries.

Original entry on oeis.org

1, 2, 4, 5, 10, 12, 14, 15, 30, 35, 40, 42, 47, 49, 51, 52, 104, 119, 134, 139, 154, 159, 164, 166, 181, 186, 191, 193, 198, 200, 202, 203, 406, 458, 510, 525, 577, 592, 607, 612, 664, 679, 694, 699, 714, 719, 724, 726, 778, 793, 808, 813, 828, 833, 838, 840

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. For example, the binary representations of {2,5,8} are:
  2 =   10,
  5 =  101,
  8 = 1000,
and since there are no columns with more than one 1, {2,5,8} is counted under a(8).

			The a(1) = 1 through a(7) = 15 subsets:
  {}   {}     {}     {}       {}       {}       {}
  {1}  {1}    {1}    {1}      {1}      {1}      {1}
       {2}    {2}    {2}      {2}      {2}      {2}
       {1,2}  {3}    {3}      {3}      {3}      {3}
              {1,2}  {4}      {4}      {4}      {4}
                     {1,2}    {5}      {5}      {5}
                     {1,4}    {1,2}    {6}      {6}
                     {2,4}    {1,4}    {1,2}    {7}
                     {3,4}    {2,4}    {1,4}    {1,2}
                     {1,2,4}  {2,5}    {1,6}    {1,4}
                              {3,4}    {2,4}    {1,6}
                              {1,2,4}  {2,5}    {2,4}
                                       {3,4}    {2,5}
                                       {1,2,4}  {3,4}
                                                {1,2,4}

Alois P. Heinz, Table of n, a(n) for n = 0..16383

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

Cf. A325094, A325096, A325097, A325100, A325103, A325104, A325105.

b:= proc(n, t) option remember; `if`(n=0, 1, b(n-1, t)+
     `if`(Bits[And](n, t)=0, b(n-1, Bits[Or](n, t)), 0))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..63);  # 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[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,10}]

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

a(16)-a(55) from Alois P. Heinz, Mar 28 2019

A325100 Heinz numbers of strict integer partitions with no binary carries.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 41, 42, 43, 47, 53, 57, 58, 59, 61, 67, 69, 71, 73, 74, 79, 83, 86, 89, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 122, 123, 127, 131, 133, 137, 139, 142, 149, 151, 157, 158, 159

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.

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 have no carries. 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}
   6: {1,2}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  41: {13}
  42: {1,2,4}

Cf. A050315, A056239, A080572, A112798, A247935, A267610.

Cf. A325095, A325096, A325097, A325100, A325101, A325103, A325110, A325119.

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}];
Select[Range[100],SquareFreeQ[#]&&stableQ[PrimePi/@First/@FactorInteger[#],Intersection[binpos[#1],binpos[#2]]!={}&]&]

A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 55, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181

Offset: 1

Gus Wiseman, Mar 30 2024

nonn

Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A binary carry of two positive integers is an overlap of binary indices. A multiset is said to be binary carry-connected iff the graph whose vertices are the elements and whose edges are binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The terms together with their prime indices begin:
     2: {1}            37: {12}              97: {25}
     3: {2}            41: {13}             101: {26}
     4: {1,1}          43: {14}             103: {27}
     5: {3}            47: {15}             107: {28}
     7: {4}            49: {4,4}            109: {29}
     8: {1,1,1}        53: {16}             113: {30}
     9: {2,2}          55: {3,5}            115: {3,9}
    11: {5}            59: {17}             121: {5,5}
    13: {6}            61: {18}             125: {3,3,3}
    16: {1,1,1,1}      64: {1,1,1,1,1,1}    127: {31}
    17: {7}            65: {3,6}            128: {1,1,1,1,1,1,1}
    19: {8}            67: {19}             131: {32}
    23: {9}            71: {20}             137: {33}
    25: {3,3}          73: {21}             139: {34}
    27: {2,2,2}        79: {22}             143: {5,6}
    29: {10}           81: {2,2,2,2}        145: {3,10}
    31: {11}           83: {23}             149: {35}
    32: {1,1,1,1,1}    89: {24}             151: {36}

Contains all powers of primes A000961 except 1.
Case of A325118 (counted by A325098) without binary containments.
For binary indices of binary indices we have A326750 = A326704 /\ A326749.
For prime indices of prime indices we have A329559 = A305078 /\ A316476.
An opposite version is A371294 = A087086 /\ A371291.
Partitions of this type are counted by A371446.
Carry-connected case of A371455 (counted by A325109).
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]

Intersection of A371455 and A325118.

A371451 Number of connected components of the binary indices of the prime indices of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2

Offset: 1

Gus Wiseman, Apr 01 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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 binary indices of prime indices of 805 are {{1,2},{3},{1,4}}, with 2 connected components {{1,2},{1,4}} and {{3}}, so a(805) = 2.

For prime indices of prime indices we have A305079, ones A305078.
Positions of ones are A325118.
Positions of first appearances are A325782.
For prime indices of binary indices we have A371452, ones A371291.
For binary indices of binary indices we have A326753, ones A326749.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A326964 counts connected set-systems, covering A323818.
Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[csm[bix/@prix[n]]],{n,100}]

zero_first_elem_and_bitmask_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(0!=bitand(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };
A371451(n) = if(1==n, 0, my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_bitmask_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Jan 29 2025

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 5, 4, 4, 4, 8, 4, 7, 7, 12, 10, 14, 12, 15, 19, 19, 21, 32, 27, 33, 40, 46, 47, 61, 52, 75, 89, 95, 104, 129, 129, 149, 176, 188, 208, 249, 257, 296, 341, 373, 394, 476, 496, 552

Offset: 0

Gus Wiseman, Apr 02 2024

These partitions are ranked by A371445.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A binary carry of two positive integers is an overlap of binary indices. An integer partition is binary carry-connected iff the graph with one vertex for each part and edges corresponding to binary carries is connected.
A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.

			The a(12) = 8 through a(14) = 7 partitions:
  (12)             (13)                         (14)
  (6,6)            (10,3)                       (7,7)
  (9,3)            (5,5,3)                      (9,5)
  (4,4,4)          (1,1,1,1,1,1,1,1,1,1,1,1,1)  (6,5,3)
  (6,3,3)                                       (5,3,3,3)
  (3,3,3,3)                                     (2,2,2,2,2,2,2)
  (2,2,2,2,2,2)                                 (1,1,1,1,1,1,1,1,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1)

The first condition (carry-connected) is A325098.
The second condition (stable) is A325109.
Ranks for binary indices of binary indices are A326750 = A326704 /\ A326749.
Ranks for prime indices of prime indices are A329559 = A305078 /\ A316476.
Ranks for prime indices of binary indices are A371294 = A087086 /\ A371291.
Ranks for binary indices of prime indices are A371445 = A325118 /\ A371455.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A326964 counts connected set-systems, covering A323818.
Cf. A019565, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[IntegerPartitions[n], stableQ[bix/@Union[#],SubsetQ]&&Length[csm[bix/@#]]<=1&]],{n,0,30}]

A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 24, 25, 26, 30, 32, 33, 34, 35, 36, 40, 42, 45, 47, 48, 50, 51, 52, 54, 55, 60, 64, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 86, 90, 94, 96, 99, 100, 102, 104, 105, 108, 110, 119, 120, 123, 125, 126, 127, 128, 130

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  18: {{1},{2},{2}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  25: {{1,2},{1,2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}

For prime indices of prime indices we have A320456.
For binary indices of binary indices we have A326754.
An opposite version is A371292, A371293.
The case with squarefree product of prime indices is A371448.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],normQ[Join@@bpe/@prix[#]]&]

A371448 Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 26, 30, 32, 33, 34, 40, 47, 48, 51, 52, 55, 60, 64, 66, 68, 80, 85, 86, 94, 96, 102, 104, 110, 120, 123, 127, 128, 132, 136, 141, 143, 160, 165, 170, 172, 187, 188, 192, 204, 205, 208, 215, 220, 221, 226, 240, 246

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have (1) squarefree product and (2) binary indices covering an initial interval.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}
  33: {{2},{1,3}}
  34: {{1},{1,2,3}}
  40: {{1},{1},{1},{1,2}}
  47: {{1,2,3,4}}
  48: {{1},{1},{1},{1},{2}}
  51: {{2},{1,2,3}}

An opposite version is A371293, A371292.
Without the squarefree condition we have A371447, see also A320456, A326754.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000], SquareFreeQ[Times@@prix[#]]&&normQ[Join@@bpe/@prix[#]]&]

Intersection of A302505 and A371447.

A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Apr 01 2024

nonn

In an antichain of sets, no edge is a proper subset of any other.

			The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.

Jakub Buczak, Table of n, a(n) for n = 1..10000

Contains all powers of primes A000961.
An opposite version is A087086, carry-connected case A371294.
For prime indices of prime indices we have A316476, carry-connected A329559.
These antichains are counted by A325109.
For binary indices of binary indices we have A326704, carry-conn. A326750.
The carry-connected case is A371445, counted by A371446.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A050320 counts set multipartitions of prime indices, see also A318360.
A070939 gives length of binary expansion.
A089259 counts set multipartitions of integer partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A116540 counts normal set multipartitions.
A302478 ranks set multipartitions, cf. A073576.
A325118 ranks carry-connected partitions, counted by A325098.
A371451 counts carry-connected components of binary indices.
Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],stableQ[bix/@prix[#],SubsetQ]&]

cp's OEIS Frontend

A247935 Number of integer partitions of n whose distinct parts have no binary carries.

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A371452 Number of connected components of the prime indices of the binary indices of n.

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A325095 Number of subsets of {1...n} with no binary carries.

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A325100 Heinz numbers of strict integer partitions with no binary carries.

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A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

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A371451 Number of connected components of the binary indices of the prime indices of n.

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A371446 Number of carry-connected integer partitions whose distinct parts have no binary containments.

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A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

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