A067368 a(n) is the smallest positive even integer that cannot be expressed as the product of two or three previous terms (not necessarily distinct).

2, 6, 10, 14, 16, 18, 22, 26, 30, 34, 38, 42, 46, 48, 50, 54, 58, 62, 66, 70, 74, 78, 80, 82, 86, 90, 94, 98, 102, 106, 110, 112, 114, 118, 122, 126, 128, 130, 134, 138, 142, 144, 146, 150, 154, 158, 162, 166, 170, 174, 176, 178, 182, 186, 190, 194, 198, 202, 206

Offset: 1

Author

Jeremiah K. Hower (jhower(AT)vt.edu), Jan 20 2002

Keywords

• nonn
• easy

Comments

a(n+1) - a(n) = 2 or 4 for all n >= 1. See A067395 for the sequence of differences.

From Jianing Song, Sep 21 2018: (Start)

Numbers of the form 2^(3t+1)*s where s is an odd number.

Also positions of 1 in A191255. (End)

The asymptotic density of this sequence is 2/7. - Amiram Eldar, May 31 2024

Examples

			8 = 2*2*2, but 10 = 2*5 cannot be expressed with factors 2 and 6, so a(3) = 10.
		

Links

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

Crossrefs

Cf. A067395, A067396, A191255, A191257, A213258.

Programs

Maple

N:= 1000:
A:= {seq(seq(2^(3*k+1)*s,s=1..N/2^(3*k+1),2),k=0..floor(log[2](N/2)/3))}:
sort(convert(A,list)); # Robert Israel, Jul 23 2019

Mathematica

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3}, 3 -> {0, 1}}] &, {0}, 9] (* A191255 *)
Flatten[Position[t, 0]] (* A005408, the odds *)
a = Flatten[Position[t, 1]] (* this sequence *)
b = Flatten[Position[t, 2]] (* A213258 *)
a/2  (* A191257 *)
b/4  (* a/2 *)
(* Clark Kimberling, May 28 2011 *)

PARI

isok(n) = valuation(n, 2)%3==1; \\ Altug Alkan, Sep 21 2018

Python

def A067368(n):
    def f(x): return n+x-sum(((x>>i)-1>>1)+1 for i in range(0,x.bit_length(),3))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m<<1 # Chai Wah Wu, Feb 17 2025

Formula

Conjecture: a(n) = a(n-1) + 2 if (n = 2a(k) + k + 1) or (n = 2a(k) + k) for some k, otherwise a(n) = a(n-1) + 4. This has been confirmed for several hundred terms.

The above conjecture is correct because there are 2*(a(k+1)-a(k)) terms that are not divisible by 4 in the k-th interval which are determined by terms that are divisible by 4. For example, there are 2*(a(2)-a(1)) = 2*(6-2) = 8 terms between a(5) = 16 and a(14) = 48 because numbers of the form 2*s are always terms where s is an odd number. So first differences of a(n) determine the corresponding intervals and the formula above always holds. - Altug Alkan, Sep 24 2018

a(n) = 2*A191257(n) = A213258(n)/2. - Jianing Song, Sep 21 2018

Extensions

Edited by John W. Layman, Jan 23 2002