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

1,1

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)

Links

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Intersection of A005843 and A003159.

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

Sequence in context: A267958 A077770 A310566 * A081523 A310567 A310568

Adjacent sequences: A108266 A108267 A108268 * A108270 A108271 A108272

Keyword

nonn,easy

Author

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

Extensions

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

More terms from Amiram Eldar, Jan 12 2021

Status

approved