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A371292 Numbers whose binary indices have prime indices covering an initial interval of positive integers.

%I A371292 #16 May 22 2024 02:14:12
%S A371292 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,
%T A371292 37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,
%U A371292 60,61,62,63,86,87,92,93,94,95,112,113,114,115,116,117,118,119
%N A371292 Numbers whose binary indices have prime indices covering an initial interval of positive integers.
%C A371292 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.
%C A371292 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.
%H A371292 John Tyler Rascoe, <a href="/A371292/b371292.txt">Table of n, a(n) for n = 0..10000</a>
%e A371292 The terms together with their prime indices of binary indices begin:
%e A371292    0: {}
%e A371292    1: {{}}
%e A371292    2: {{1}}
%e A371292    3: {{},{1}}
%e A371292    6: {{1},{2}}
%e A371292    7: {{},{1},{2}}
%e A371292    8: {{1,1}}
%e A371292    9: {{},{1,1}}
%e A371292   10: {{1},{1,1}}
%e A371292   11: {{},{1},{1,1}}
%e A371292   12: {{2},{1,1}}
%e A371292   13: {{},{2},{1,1}}
%e A371292   14: {{1},{2},{1,1}}
%e A371292   15: {{},{1},{2},{1,1}}
%e A371292   22: {{1},{2},{3}}
%e A371292   23: {{},{1},{2},{3}}
%e A371292   28: {{2},{1,1},{3}}
%e A371292   29: {{},{2},{1,1},{3}}
%e A371292   30: {{1},{2},{1,1},{3}}
%e A371292   31: {{},{1},{2},{1,1},{3}}
%e A371292   32: {{1,2}}
%t A371292 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A371292 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371292 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371292 Select[Range[0,100],normQ[Join@@prix/@bpe[#]]&]
%o A371292 (Python)
%o A371292 from itertools import count, islice
%o A371292 from sympy import sieve, factorint
%o A371292 def a_gen():
%o A371292     for n in count(0):
%o A371292         s = set()
%o A371292         b = [(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1']
%o A371292         for i in b:
%o A371292             p = factorint(i)
%o A371292             for j in p:
%o A371292                 s.add(sieve.search(j)[0])
%o A371292         x = sorted(s)
%o A371292         y = len(x)
%o A371292         if sum(x) == (y*(y+1))//2:
%o A371292             yield n
%o A371292 A371292_list = list(islice(a_gen(), 65)) # _John Tyler Rascoe_, May 21 2024
%Y A371292 The case with squarefree product of prime indices is A371293.
%Y A371292 For binary indices of each prime index we have A371447, A371448.
%Y A371292 The connected components of this multiset system are counted by A371452.
%Y A371292 A000009 counts partitions covering initial interval, compositions A107429.
%Y A371292 A000670 counts patterns, ranked by A333217.
%Y A371292 A011782 counts multisets covering an initial interval.
%Y A371292 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A371292 A070939 gives length of binary expansion.
%Y A371292 A131689 counts patterns by number of distinct parts.
%Y A371292 Cf. A000040, A001222, A055887, A255906, A326782, A368109, A371291, A371294.
%K A371292 nonn,base
%O A371292 0,3
%A A371292 _Gus Wiseman_, Mar 27 2024