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A234000
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Numbers of the form 4^i*(8*j+1).
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12
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1, 4, 9, 16, 17, 25, 33, 36, 41, 49, 57, 64, 65, 68, 73, 81, 89, 97, 100, 105, 113, 121, 129, 132, 137, 144, 145, 153, 161, 164, 169, 177, 185, 193, 196, 201, 209, 217, 225, 228, 233, 241, 249, 256, 257, 260, 265, 272, 273, 281, 289, 292, 297, 305, 313, 321, 324, 329, 337, 345
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OFFSET
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1,2
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COMMENTS
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Closed under multiplication.
Contains all even powers of primes.
A subgroup of the positive integers under the binary operation A059897(.,.). For all n, a(n) has no Fermi-Dirac factor of 2 and if m_k denotes the number of Fermi-Dirac factors of a(n) that are congruent to k modulo 8, m_3, m_5 and m_7 have the same parity. It can further be shown (1) all numbers that meet these requirements are in the sequence and (2) this implies closure under A059897(.,.).
(End)
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LINKS
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FORMULA
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MAPLE
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N:= 1000: # to get all terms <= N
{seq(seq(4^i*(8*k+1), k = 0 .. floor((N * 4^(-i)-1)/8)), i=0..floor(log[4](N)))}; # Robert Israel, Aug 26 2014
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PROG
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(PARI) list(lim)=my(v=List(), t); for(e=0, logint(lim\1, 4), t=4^e; forstep(k=t, lim, 8*t, listput(v, k))); Set(v) \\ Charles R Greathouse IV, Jan 12 2017
(Python)
from itertools import count, islice
def A234000_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==1, count(max(startvalue, 1)))
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CROSSREFS
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Cf. A055046 (Numbers of the form 4^i*(8*j+3)).
Cf. A055045 (Numbers of the form 4^i*(8*j+5)).
Cf. A004215 (Numbers of the form 4^i*(8*j+7)).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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