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

Author

V. Raman, Dec 18 2013

Keywords

• nonn
• easy

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

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, A055044.

Programs

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

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

Python

def A234000(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>(i<<1))-1>>3)+1 for i in range((x.bit_length()>>1)+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

Formula

a(n) = 6n + O(log n). - Charles R Greathouse IV, Dec 19 2013

a(n) = A055044(n)/2. - Chai Wah Wu, Mar 19 2025