A108269 Numbers of the form (2*m - 1)*4^k where m >= 1, k >= 1.

4, 12, 16, 20, 28, 36, 44, 48, 52, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 140, 144, 148, 156, 164, 172, 176, 180, 188, 192, 196, 204, 208, 212, 220, 228, 236, 240, 244, 252, 256, 260, 268, 272, 276, 284, 292, 300, 304, 308, 316, 320, 324, 332

Offset: 1

Author

Andras Erszegi (erszegi.andras(AT)chello.hu), May 30 2005

Keywords

• nonn
• easy

Comments

Numbers of terms in nonnegative integer sequences the sum of which is never a square.

The sum of a sequence of consecutive nonnegative integers starting with k is never a square for any k, if and only if the number of the terms in the sequence can be expressed as (2*m - 1) * 2^(2*n), m and n being any positive integers. (Proved by Alfred Vella, Jun 14 2005.)

Odious and evil terms alternate. - Vladimir Shevelev, Jun 22 2009

Even numbers whose binary representation ends in an even number of zeros. - Amiram Eldar, Jan 12 2021

From Antti Karttunen, Jan 28 2023: (Start)

Numbers k for which the parity of k is equal to that of A048675(k).

A multiplicative semigroup; if m and n are in the sequence then so is m*n. (End)

Examples

			a( 1, 1 ) = 4, a( 2, 1) = 12, etc.
For a( 1, 1 ): the sum of 4 consecutive nonnegative integers (4k+6, if the first term is k) is never a square.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005843 and A003159.

Cf. A000069, A001969, A017113 (primitive terms), A036554, A328981 (characteristic function), A359794 (complement).

Programs

Mathematica

Select[2 * Range[200], EvenQ @ IntegerExponent[#, 2] &] (* Amiram Eldar, Jan 12 2021 *)

PARI

is(n)=my(e=valuation(n,2)); e>1 && e%2==0 \\ Charles R Greathouse IV, Nov 03 2016

Python

def A108269(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):
        c, s = n+(x+1>>1), bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c += int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 29 2025

Formula

a(n) = 6*n + O(log n). - Charles R Greathouse IV, Nov 03 2016 [Corrected by Amiram Eldar, Jan 12 2021]

a(n) = 2 * A036554(n) = 4 * A003159(n). - Amiram Eldar, Jan 12 2021

Extensions

Entry revised by N. J. A. Sloane, Jun 26 2005

More terms from Amiram Eldar, Jan 12 2021