Displaying 1-10 of 14 results found.
Numbers whose binary indices are connected, where two numbers are connected iff they have a common factor.
+10
9
1, 2, 4, 8, 10, 16, 32, 34, 36, 38, 40, 42, 44, 46, 64, 128, 130, 136, 138, 160, 162, 164, 166, 168, 170, 172, 174, 256, 260, 288, 290, 292, 294, 296, 298, 300, 302, 416, 418, 420, 422, 424, 426, 428, 430, 512, 514, 520, 522, 528, 530, 536, 538, 544, 546, 548
COMMENTS
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 empty set is not considered connected.
EXAMPLE
The terms together with their binary expansions and binary indices begin:
1: 1 ~ {1}
2: 10 ~ {2}
4: 100 ~ {3}
8: 1000 ~ {4}
10: 1010 ~ {2,4}
16: 10000 ~ {5}
32: 100000 ~ {6}
34: 100010 ~ {2,6}
36: 100100 ~ {3,6}
38: 100110 ~ {2,3,6}
40: 101000 ~ {4,6}
42: 101010 ~ {2,4,6}
44: 101100 ~ {3,4,6}
46: 101110 ~ {2,3,4,6}
64: 1000000 ~ {7}
128: 10000000 ~ {8}
130: 10000010 ~ {2,8}
136: 10001000 ~ {4,8}
138: 10001010 ~ {2,4,8}
160: 10100000 ~ {6,8}
162: 10100010 ~ {2,6,8}
164: 10100100 ~ {3,6,8}
MATHEMATICA
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]]]]]]]]];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 1000], Length[csm[prix/@bpe[#]]]==1&]
CROSSREFS
For prime indices of each prime index we have A305078. For binary indices of each binary index we have A326749. A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
A087086 lists numbers whose binary indices are pairwise indivisible.
A096111 gives product of binary indices.
Numbers whose binary indices have prime indices covering an initial interval of positive integers.
+10
8
0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 86, 87, 92, 93, 94, 95, 112, 113, 114, 115, 116, 117, 118, 119
COMMENTS
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.
EXAMPLE
The terms together with their prime indices of binary indices begin:
0: {}
1: {{}}
2: {{1}}
3: {{},{1}}
6: {{1},{2}}
7: {{},{1},{2}}
8: {{1,1}}
9: {{},{1,1}}
10: {{1},{1,1}}
11: {{},{1},{1,1}}
12: {{2},{1,1}}
13: {{},{2},{1,1}}
14: {{1},{2},{1,1}}
15: {{},{1},{2},{1,1}}
22: {{1},{2},{3}}
23: {{},{1},{2},{3}}
28: {{2},{1,1},{3}}
29: {{},{2},{1,1},{3}}
30: {{1},{2},{1,1},{3}}
31: {{},{1},{2},{1,1},{3}}
32: {{1,2}}
MATHEMATICA
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], normQ[Join@@prix/@bpe[#]]&]
PROG
(Python)
from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
for n in count(0):
s = set()
b = [(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1']
for i in b:
p = factorint(i)
for j in p:
s.add(sieve.search(j)[0])
x = sorted(s)
y = len(x)
if sum(x) == (y*(y+1))//2:
yield n
CROSSREFS
The case with squarefree product of prime indices is A371293. The connected components of this multiset system are counted by A371452. A000009 counts partitions covering initial interval, compositions A107429.
A011782 counts multisets covering an initial interval.
A070939 gives length of binary expansion.
A131689 counts patterns by number of distinct parts.
Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.
+10
8
1, 2, 4, 8, 16, 32, 40, 64, 128, 160, 256, 288, 296, 416, 512, 520, 544, 552, 640, 672, 800, 808, 928, 1024, 2048, 2176, 2304, 2432, 2560, 2688, 2816, 2944, 4096, 8192, 8200, 8224, 8232, 8320, 8352, 8480, 8488, 8608, 8704, 8712, 8736, 8744, 8832, 8864, 8992
COMMENTS
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.
EXAMPLE
The terms together with their prime indices of binary indices begin:
1: {{}}
2: {{1}}
4: {{2}}
8: {{1,1}}
16: {{3}}
32: {{1,2}}
40: {{1,1},{1,2}}
64: {{4}}
128: {{1,1,1}}
160: {{1,2},{1,1,1}}
256: {{2,2}}
288: {{1,2},{2,2}}
296: {{1,1},{1,2},{2,2}}
416: {{1,2},{1,1,1},{2,2}}
512: {{1,3}}
520: {{1,1},{1,3}}
544: {{1,2},{1,3}}
552: {{1,1},{1,2},{1,3}}
640: {{1,1,1},{1,3}}
672: {{1,2},{1,1,1},{1,3}}
800: {{1,2},{2,2},{1,3}}
808: {{1,1},{1,2},{2,2},{1,3}}
928: {{1,2},{1,1,1},{2,2},{1,3}}
MATHEMATICA
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}]]]];
Select[Range[1000], stableQ[bpe[#], Divisible]&&connectedQ[prix/@bpe[#]]&]
CROSSREFS
For binary indices of binary indices we have A326750, non-primitive A326749. For prime indices of prime indices we have A329559, non-primitive A305078. For binary indices of prime indices we have A371445, non-primitive A325118. A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A051026, A285572, A303362, A304713, A305079, A316476, A319496, A319719, A326704, A371446.
Numbers whose distinct prime indices are binary carry-connected and have no binary containments.
+10
6
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
COMMENTS
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.
EXAMPLE
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}
MATHEMATICA
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&]
CROSSREFS
Contains all powers of primes A000961 except 1. Partitions of this type are counted by A371446. A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.
Numbers whose binary indices have squarefree product.
+10
5
0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 48, 49, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 112, 113, 512, 513, 516, 517, 576, 577, 580, 581, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1040, 1041, 1042
COMMENTS
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.
EXAMPLE
The terms together with their binary expansions and binary indices begin:
0: 0 ~ {}
1: 1 ~ {1}
2: 10 ~ {2}
3: 11 ~ {1,2}
4: 100 ~ {3}
5: 101 ~ {1,3}
6: 110 ~ {2,3}
7: 111 ~ {1,2,3}
16: 10000 ~ {5}
17: 10001 ~ {1,5}
18: 10010 ~ {2,5}
19: 10011 ~ {1,2,5}
20: 10100 ~ {3,5}
21: 10101 ~ {1,3,5}
22: 10110 ~ {2,3,5}
23: 10111 ~ {1,2,3,5}
32: 100000 ~ {6}
33: 100001 ~ {1,6}
48: 110000 ~ {5,6}
49: 110001 ~ {1,5,6}
64: 1000000 ~ {7}
65: 1000001 ~ {1,7}
66: 1000010 ~ {2,7}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], SquareFreeQ[Times@@bpe[#]]&]
CROSSREFS
For prime instead of binary indices we have A302505. A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A325118, A326782, A371290, A371291, A371292, A371293, A371443, A371446, A371448, A371449, A371452, A371453.
Number of carry-connected integer partitions whose distinct parts have no binary containments.
+10
5
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
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}]
CROSSREFS
The first condition (carry-connected) is A325098. The second condition (stable) is A325109. A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
Numbers whose binary indices of prime indices cover an initial interval of positive integers.
+10
5
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
COMMENTS
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.
EXAMPLE
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}}
MATHEMATICA
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[#]]&]
CROSSREFS
For prime indices of prime indices we have A320456. For binary indices of binary indices we have A326754. 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.
A011782 counts multisets covering an initial interval.
A070939 gives length of binary expansion.
A131689 counts patterns by number of distinct parts.
Number of connected components of the binary indices of the prime indices of n.
+10
5
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
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
Positions of first appearances are A325782. A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.
Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.
+10
4
1, 2, 3, 6, 7, 22, 23, 32, 33, 48, 49, 86, 87, 112, 113, 516, 517, 580, 581, 1110, 1111, 1136, 1137, 1604, 1605, 5206, 5207, 5232, 5233, 5700, 5701, 8212, 8213, 9236, 9237, 13332, 13333, 16386, 16387, 16450, 16451, 17474, 17475, 21570, 21571, 24576, 24577
COMMENTS
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.
EXAMPLE
The terms together with their prime indices of binary indices begin:
1: {{}}
2: {{1}}
3: {{},{1}}
6: {{1},{2}}
7: {{},{1},{2}}
22: {{1},{2},{3}}
23: {{},{1},{2},{3}}
32: {{1,2}}
33: {{},{1,2}}
48: {{3},{1,2}}
49: {{},{3},{1,2}}
86: {{1},{2},{3},{4}}
87: {{},{1},{2},{3},{4}}
112: {{3},{1,2},{4}}
113: {{},{3},{1,2},{4}}
516: {{2},{1,3}}
517: {{},{2},{1,3}}
580: {{2},{4},{1,3}}
581: {{},{2},{4},{1,3}}
MATHEMATICA
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[1000], SquareFreeQ[Times @@ bpe[#]]&&normQ[Join@@prix/@bpe[#]]&]
CROSSREFS
Without the covering condition we have A371289. Without squarefree product we have A371292. Interchanging binary and prime indices gives A371448. A000009 counts partitions covering initial interval, compositions A107429.
A011782 counts multisets covering an initial interval.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A131689 counts patterns by number of distinct parts.
A326701 lists BII-numbers of set partitions.
A368533 lists numbers with squarefree binary indices, prime indices A302478.
Numbers whose binary indices are nonprime numbers.
+10
4
1, 8, 9, 32, 33, 40, 41, 128, 129, 136, 137, 160, 161, 168, 169, 256, 257, 264, 265, 288, 289, 296, 297, 384, 385, 392, 393, 416, 417, 424, 425, 512, 513, 520, 521, 544, 545, 552, 553, 640, 641, 648, 649, 672, 673, 680, 681, 768, 769, 776, 777, 800, 801, 808
COMMENTS
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.
EXAMPLE
The terms together with their binary expansions and binary indices begin:
1: 1 ~ {1}
8: 1000 ~ {4}
9: 1001 ~ {1,4}
32: 100000 ~ {6}
33: 100001 ~ {1,6}
40: 101000 ~ {4,6}
41: 101001 ~ {1,4,6}
128: 10000000 ~ {8}
129: 10000001 ~ {1,8}
136: 10001000 ~ {4,8}
137: 10001001 ~ {1,4,8}
160: 10100000 ~ {6,8}
161: 10100001 ~ {1,6,8}
168: 10101000 ~ {4,6,8}
169: 10101001 ~ {1,4,6,8}
256: 100000000 ~ {9}
257: 100000001 ~ {1,9}
264: 100001000 ~ {4,9}
265: 100001001 ~ {1,4,9}
288: 100100000 ~ {6,9}
289: 100100001 ~ {1,6,9}
296: 100101000 ~ {4,6,9}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[100], And@@Not/@PrimeQ/@bpe[#]&]
CROSSREFS
For powers of 2 instead of nonprime numbers we have A253317. For prime indices instead of binary indices we have A320628. For prime instead of nonprime we have A326782. For composite numbers we have A371444. A070939 gives length of binary expansion.
A096111 gives product of binary indices.
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