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A371453
Numbers whose binary indices are all squarefree semiprimes.
4
32, 512, 544, 8192, 8224, 8704, 8736, 16384, 16416, 16896, 16928, 24576, 24608, 25088, 25120, 1048576, 1048608, 1049088, 1049120, 1056768, 1056800, 1057280, 1057312, 1064960, 1064992, 1065472, 1065504, 1073152, 1073184, 1073664, 1073696, 2097152, 2097184
OFFSET
1,1
COMMENTS
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.
EXAMPLE
The terms together with their binary expansions and binary indices begin:
32: 100000 ~ {6}
512: 1000000000 ~ {10}
544: 1000100000 ~ {6,10}
8192: 10000000000000 ~ {14}
8224: 10000000100000 ~ {6,14}
8704: 10001000000000 ~ {10,14}
8736: 10001000100000 ~ {6,10,14}
16384: 100000000000000 ~ {15}
16416: 100000000100000 ~ {6,15}
16896: 100001000000000 ~ {10,15}
16928: 100001000100000 ~ {6,10,15}
24576: 110000000000000 ~ {14,15}
24608: 110000000100000 ~ {6,14,15}
25088: 110001000000000 ~ {10,14,15}
25120: 110001000100000 ~ {6,10,14,15}
1048576: 100000000000000000000 ~ {21}
MAPLE
M:= 26: # for terms < 2^M
P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):
S:= select(`<`, {seq(seq(P[i]*P[j], i=1..j-1), j=1..nP)}, M+1):
R:= map(proc(s) local i; add(2^(i-1), i=s) end proc, combinat:-powerset(S) minus {{}}):
sort(convert(R, list)); # Robert Israel, Apr 04 2024
MATHEMATICA
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[10000], And@@sqfsemi/@bix[#]&]
PROG
(Python)
def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1], 1) if j=='1')
(Python)
from math import isqrt
from sympy import primepi, primerange
def A371453(n):
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)))
def A006881(n):
m, k = n, f(n, n)
while m != k:
m, k = k, f(k, n)
return m
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
CROSSREFS
Partitions of this type are counted by A002100, squarefree case of A101048.
For primes instead of squarefree semiprimes we get A326782.
For prime indices instead of binary indices we have A339113, A339112.
Allowing any squarefree numbers gives A368533.
This is the squarefree case of A371454.
A001358 lists squarefree semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Sequence in context: A022627 A238534 A297091 * A271577 A116003 A035711
KEYWORD
nonn,base
AUTHOR
Gus Wiseman, Apr 02 2024
STATUS
approved