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Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.
A binary carry of two positive integers is an overlap of binary indices. An integer partition A multiset is said to be binary carry-connected iff the graph whose vertices are the parts elements and whose edges are binary carries is connected.
For binary indices of binary indices we have A326750, subset of = A326704 /\ A326749.
For prime indices of prime indices we have A329559, subset of = A305078 /\ A316476.
An opposite version is A371294, intersection of = A087086 and /\ A371291.
Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A316476, A325097, A325105, A325107, A325119, A371452.
Partitions of this type are counted by A371446.
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Carry-connected case of A371455 (counted by A325109).
Carry-connected case of A371455 (counted by A325109).
`A051026 gives the number of primitive subsets of the integers 1 to n.
Cf. A000040, A000720, A001222, A019565, A056239, A080572, A112798, A304713, A304716, A305079, A305148, A316476, A325105, A325107, A325119, A371446, A371452.
Cf. A304713, A304716, A305079, A305148, A316476, `A325094, A325095, A325097, A325105, `A325106, A325107, A325108, A325119, `A326704, `A371291, A371452.
Numbers whose distinct prime indices are binary carry-connected and have no binary containments.
Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
A binary carry of two positive integers is an overlap of binary indices. An integer partition is binary carry-connected if iff the graph whose vertices are the parts 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.
Carry-connected case of A325097, A371455 (counted by A247935, strict A325110 strptns_wo_bincontainA325109).
Case of A325118 (counted by A325098) without binary containments, counted by A325098 ptns_bincarconn, strict A325099 strptns_bincarconn.
For binary indices of binary indices we have A326750, the case subset of A326749 without binary containments.
For prime indices of prime indices we have A329559, primitive case subset of A305078.
The An opposite version is A371294, intersection of A087086 and A371291.
`A051026 gives the number of primitive subsets of the integers 1 to n.
A087086 lists numbers with pairwise indivisible binary indices, relatively prime A328671.
A096111 gives product of binary indices.
A325109 counts partitions without binary containments.
Cf. A000040, A000720, A001222, A019565, `A050315, A056239, A080572, A112798, `A371446.
Cf. A304713, A304716, A305079, `A305148, `A316476, `A325094, A325095, `A325096, ~A325100, ~A325101, ~A325102, ~A325103, A325097, A325105, `A325106, A325107, `A325108, A325119, `A326704, `A326753, A371291, A371452.
allocated for Gus WisemanNumbers whose distinct prime indices are carry-connected and have no binary containments.
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
1,1
Also Heinz numbers of carry-connected integer partitions whose distinct parts have no binary containments. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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 if the graph whose vertices are the parts 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}
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&]
Contains all powers of primes A000961 except 1.
Carry-connected case of A325097, counted by A247935, strict A325110 strptns_wo_bincontain.
Case of A325118 without binary containments, counted by A325098 ptns_bincarconn, strict A325099 strptns_bincarconn.
For binary indices of binary indices we have A326750, the case of A326749 without binary containments.
For prime indices of prime indices we have A329559, primitive case of A305078.
The opposite version is A371294, intersection of A087086 and A371291.
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.
A051026 gives the number of primitive subsets of the integers 1 to n.
A070939 gives length of binary expansion.
A087086 lists numbers with pairwise indivisible binary indices, relatively prime A328671.
A096111 gives product of binary indices.
A325109 counts partitions without binary containments.
A326964 counts connected set-systems, covering A323818.
Cf. A000040, A000720, A001222, A019565, `A050315, A056239, A080572, A112798, `A371446.
Cf. A304713, A304716, A305079, `A305148, `A316476, `A325094, A325095, `A325096, ~A325100, ~A325101, ~A325102, ~A325103, A325105, `A325106, A325107, `A325108, A325119, `A326704, `A326753, A371452.
allocated
nonn
Gus Wiseman, Mar 30 2024
approved
editing