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%I #60 Jul 09 2022 11:06:46
%S 1,4,9,16,17,25,33,36,41,49,57,64,65,68,73,81,89,97,100,105,113,121,
%T 129,132,137,144,145,153,161,164,169,177,185,193,196,201,209,217,225,
%U 228,233,241,249,256,257,260,265,272,273,281,289,292,297,305,313,321,324,329,337,345
%N Numbers of the form 4^i*(8*j+1).
%C Squares modulo all powers of 2. - _Robert Israel_, Aug 26 2014
%C From _Peter Munn_, Dec 11 2019: (Start)
%C Closed under multiplication.
%C Contains all even powers of primes.
%C 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(.,.).
%C (End)
%H Robert Israel, <a href="/A234000/b234000.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 6n + O(log n). - _Charles R Greathouse IV_, Dec 19 2013
%p N:= 1000: # to get all terms <= N
%p {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
%o (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
%o (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
%o (Python)
%o from itertools import count, islice
%o def A234000_gen(startvalue=1): # generator of terms >= startvalue
%o return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==1,count(max(startvalue,1)))
%o A234000_list = list(islice(A234000_gen(),30)) # _Chai Wah Wu_, Jul 09 2022
%Y Cf. A055046 (Numbers of the form 4^i*(8*j+3)).
%Y Cf. A055045 (Numbers of the form 4^i*(8*j+5)).
%Y Cf. A004215 (Numbers of the form 4^i*(8*j+7)).
%Y Cf. A059897.
%K nonn,easy
%O 1,2
%A _V. Raman_, Dec 18 2013
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