login
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).
10
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
1,1
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
LINKS
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
EXAMPLE
8 = 2*2*2, but 10 = 2*5 cannot be expressed with factors 2 and 6, so a(3) = 10.
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 *)
PROG
(PARI) isok(n) = valuation(n, 2)%3==1; \\ Altug Alkan, Sep 21 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jeremiah K. Hower (jhower(AT)vt.edu), Jan 20 2002
EXTENSIONS
Edited by John W. Layman, Jan 23 2002
STATUS
approved