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