A371453 - OEIS

%I #14 Aug 16 2024 20:48:16

%S 32,512,544,8192,8224,8704,8736,16384,16416,16896,16928,24576,24608,

%T 25088,25120,1048576,1048608,1049088,1049120,1056768,1056800,1057280,

%U 1057312,1064960,1064992,1065472,1065504,1073152,1073184,1073664,1073696,2097152,2097184

%N Numbers whose binary indices are all squarefree semiprimes.

%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.

%e The terms together with their binary expansions and binary indices begin:

%e        32:                 100000 ~ {6}

%e       512:             1000000000 ~ {10}

%e       544:             1000100000 ~ {6,10}

%e      8192:         10000000000000 ~ {14}

%e      8224:         10000000100000 ~ {6,14}

%e      8704:         10001000000000 ~ {10,14}

%e      8736:         10001000100000 ~ {6,10,14}

%e     16384:        100000000000000 ~ {15}

%e     16416:        100000000100000 ~ {6,15}

%e     16896:        100001000000000 ~ {10,15}

%e     16928:        100001000100000 ~ {6,10,15}

%e     24576:        110000000000000 ~ {14,15}

%e     24608:        110000000100000 ~ {6,14,15}

%e     25088:        110001000000000 ~ {10,14,15}

%e     25120:        110001000100000 ~ {6,10,14,15}

%e   1048576:  100000000000000000000 ~ {21}

%p M:= 26: # for terms < 2^M

%p P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):

%p S:= select(`<`,{seq(seq(P[i]*P[j],i=1..j-1),j=1..nP)},M+1):

%p R:= map(proc(s) local i; add(2^(i-1),i=s) end proc, combinat:-powerset(S) minus {{}}):

%p sort(convert(R,list)); # _Robert Israel_, Apr 04 2024

%t bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;

%t Select[Range[10000],And@@sqfsemi/@bix[#]&]

%o (Python)

%o def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange

%o def A371453(n):

%o     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     def A006881(n):

%o         m, k = n, f(n,n)

%o         while m != k:

%o             m, k = k, f(k,n)

%o         return m

%o     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 Partitions of this type are counted by A002100, squarefree case of A101048.

%Y For primes instead of squarefree semiprimes we get A326782.

%Y For prime indices instead of binary indices we have A339113, A339112.

%Y Allowing any squarefree numbers gives A368533.

%Y This is the squarefree case of A371454.

%Y A001358 lists squarefree semiprimes, squarefree A006881.

%Y A005117 lists squarefree numbers.

%Y A048793 lists binary indices, reverse A272020, length A000120, sum A029931.

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y Cf. A087086, A112798, A296119, A302478, A326031, A367905, A368109, A371450.

%K nonn,base

%O 1,1

%A _Gus Wiseman_, Apr 02 2024