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A371292
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Numbers whose binary indices have prime indices covering an initial interval of positive integers.
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8
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}}
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MATHEMATICA
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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[#]]&]
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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