A234000 Numbers of the form 4^i*(8*j+1).

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

Adjacent sequences: A233997 A233998 A233999 * A234001 A234002 A234003

Keyword

nonn,easy

Author

V. Raman, Dec 18 2013

Status

approved