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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). 5
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 (list; graph; refs; listen; history; text; internal format)
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)

LINKS

Table of n, a(n) for n=1..59.

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.

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

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

Sequence in context: A302797 A130319 A191256 * A191259 A184914 A232176

Adjacent sequences:  A067365 A067366 A067367 * A067369 A067370 A067371

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

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Last modified June 19 11:23 EDT 2019. Contains 324219 sequences. (Running on oeis4.)