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A234000
Numbers of the form 4^i*(8*j+1).
12
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
OFFSET
1,2
COMMENTS
Squares modulo all powers of 2. - Robert Israel, Aug 26 2014
From Peter Munn, Dec 11 2019: (Start)
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)
LINKS
FORMULA
a(n) = 6n + O(log n). - Charles R Greathouse IV, Dec 19 2013
MAPLE
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
PROG
(PARI) is_A234000(n)=(n/4^valuation(n, 4))%8==1 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013; minor improvement by M. F. Hasler, Jan 02 2014
(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)))
A234000_list = list(islice(A234000_gen(), 30)) # Chai Wah Wu, Jul 09 2022
CROSSREFS
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)).
Cf. A059897.
Sequence in context: A313316 A010393 A010425 * A313317 A292674 A313318
KEYWORD
nonn,easy
AUTHOR
V. Raman, Dec 18 2013
STATUS
approved