cp's OEIS Frontend

A371453 Numbers whose binary indices are all squarefree semiprimes.

%I A371453 #14 Aug 16 2024 20:48:16
%S A371453 32,512,544,8192,8224,8704,8736,16384,16416,16896,16928,24576,24608,
%T A371453 25088,25120,1048576,1048608,1049088,1049120,1056768,1056800,1057280,
%U A371453 1057312,1064960,1064992,1065472,1065504,1073152,1073184,1073664,1073696,2097152,2097184
%N A371453 Numbers whose binary indices are all squarefree semiprimes.
%C A371453 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.
%e A371453 The terms together with their binary expansions and binary indices begin:
%e A371453        32:                 100000 ~ {6}
%e A371453       512:             1000000000 ~ {10}
%e A371453       544:             1000100000 ~ {6,10}
%e A371453      8192:         10000000000000 ~ {14}
%e A371453      8224:         10000000100000 ~ {6,14}
%e A371453      8704:         10001000000000 ~ {10,14}
%e A371453      8736:         10001000100000 ~ {6,10,14}
%e A371453     16384:        100000000000000 ~ {15}
%e A371453     16416:        100000000100000 ~ {6,15}
%e A371453     16896:        100001000000000 ~ {10,15}
%e A371453     16928:        100001000100000 ~ {6,10,15}
%e A371453     24576:        110000000000000 ~ {14,15}
%e A371453     24608:        110000000100000 ~ {6,14,15}
%e A371453     25088:        110001000000000 ~ {10,14,15}
%e A371453     25120:        110001000100000 ~ {6,10,14,15}
%e A371453   1048576:  100000000000000000000 ~ {21}
%p A371453 M:= 26: # for terms < 2^M
%p A371453 P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):
%p A371453 S:= select(`<`,{seq(seq(P[i]*P[j],i=1..j-1),j=1..nP)},M+1):
%p A371453 R:= map(proc(s) local i; add(2^(i-1),i=s) end proc, combinat:-powerset(S) minus {{}}):
%p A371453 sort(convert(R,list)); # _Robert Israel_, Apr 04 2024
%t A371453 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371453 sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
%t A371453 Select[Range[10000],And@@sqfsemi/@bix[#]&]
%o A371453 (Python)
%o A371453 def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')
%o A371453 (Python)
%o A371453 from math import isqrt
%o A371453 from sympy import primepi, primerange
%o A371453 def A371453(n):
%o A371453     def f(x,n): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
%o A371453     def A006881(n):
%o A371453         m, k = n, f(n,n)
%o A371453         while m != k:
%o A371453             m, k = k, f(k,n)
%o A371453         return m
%o A371453     return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # _Chai Wah Wu_, Aug 16 2024
%Y A371453 Partitions of this type are counted by A002100, squarefree case of A101048.
%Y A371453 For primes instead of squarefree semiprimes we get A326782.
%Y A371453 For prime indices instead of binary indices we have A339113, A339112.
%Y A371453 Allowing any squarefree numbers gives A368533.
%Y A371453 This is the squarefree case of A371454.
%Y A371453 A001358 lists squarefree semiprimes, squarefree A006881.
%Y A371453 A005117 lists squarefree numbers.
%Y A371453 A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A371453 A070939 gives length of binary expansion.
%Y A371453 A096111 gives product of binary indices.
%Y A371453 Cf. A087086, A112798, A296119, A302478, A326031, A367905, A368109, A371450.
%K A371453 nonn,base
%O A371453 1,1
%A A371453 _Gus Wiseman_, Apr 02 2024