A371453 - OEIS

A371453

Numbers whose binary indices are all squarefree semiprimes.

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

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.

LINKS

Table of n, a(n) for n=1..33.

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.

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

Sequence in context: A022627 A238534 A297091 * A271577 A116003 A035711

Adjacent sequences: A371450 A371451 A371452 * A371454 A371455 A371456

KEYWORD

nonn,base

AUTHOR

Gus Wiseman, Apr 02 2024

STATUS

approved