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A371453
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Numbers whose binary indices are all squarefree semiprimes.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}
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MAPLE
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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 {{}}):
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MATHEMATICA
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bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[10000], And@@sqfsemi/@bix[#]&]
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PROG
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(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 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)))
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
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CROSSREFS
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Partitions of this type are counted by A002100, squarefree case of A101048.
For primes instead of squarefree semiprimes we get A326782.
Allowing any squarefree numbers gives A368533.
This is the squarefree case of A371454.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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