cp's OEIS Frontend

A371454 Numbers whose binary indices are all semiprimes.

%I A371454 #11 Aug 16 2024 20:48:24
%S A371454 8,32,40,256,264,288,296,512,520,544,552,768,776,800,808,8192,8200,
%T A371454 8224,8232,8448,8456,8480,8488,8704,8712,8736,8744,8960,8968,8992,
%U A371454 9000,16384,16392,16416,16424,16640,16648,16672,16680,16896,16904,16928,16936,17152
%N A371454 Numbers whose binary indices are all semiprimes.
%C A371454 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 A371454 The terms together with their binary expansions and binary indices begin:
%e A371454      8:           1000 ~ {4}
%e A371454     32:         100000 ~ {6}
%e A371454     40:         101000 ~ {4,6}
%e A371454    256:      100000000 ~ {9}
%e A371454    264:      100001000 ~ {4,9}
%e A371454    288:      100100000 ~ {6,9}
%e A371454    296:      100101000 ~ {4,6,9}
%e A371454    512:     1000000000 ~ {10}
%e A371454    520:     1000001000 ~ {4,10}
%e A371454    544:     1000100000 ~ {6,10}
%e A371454    552:     1000101000 ~ {4,6,10}
%e A371454    768:     1100000000 ~ {9,10}
%e A371454    776:     1100001000 ~ {4,9,10}
%e A371454    800:     1100100000 ~ {6,9,10}
%e A371454    808:     1100101000 ~ {4,6,9,10}
%t A371454 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371454 semi[n_]:=PrimeOmega[n]==2;
%t A371454 Select[Range[10000],And@@semi/@bix[#]&]
%o A371454 (Python)
%o A371454 from math import isqrt
%o A371454 from sympy import primepi, primerange
%o A371454 def A371454(n):
%o A371454     def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
%o A371454     def A001358(n):
%o A371454         m, k = n, f(n,n)
%o A371454         while m != k:
%o A371454             m, k = k, f(k,n)
%o A371454         return m
%o A371454     return sum(1<<A001358(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # _Chai Wah Wu_, Aug 16 2024
%Y A371454 Partitions of this type are counted by A101048, squarefree case A002100.
%Y A371454 For primes instead of semiprimes we get A326782.
%Y A371454 For prime indices instead of binary indices we have A339112, A339113.
%Y A371454 The squarefree case is A371453.
%Y A371454 A001358 lists semiprimes, squarefree A006881.
%Y A371454 A005117 lists squarefree numbers.
%Y A371454 A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A371454 A070939 gives length of binary expansion.
%Y A371454 A096111 gives product of binary indices.
%Y A371454 Cf. A087086, A296119, A302478, A326031, A367905, A368109, A368533, A371450.
%K A371454 nonn,base
%O A371454 1,1
%A A371454 _Gus Wiseman_, Apr 02 2024