1,2

Charles R Greathouse IV asked for a proof showing that this sequence is infinite (SeqFan mailing list, Jan 2014).

Significant jumps occur at a(3)=4=2*a(2), a(5)=10=2*a(4), a(9)=28=2*a(8), a(17)=82=2*a(16), a(31)=236 >> a(29)=122, a(47)=628 >> a(44)=302, a(70)=1622 >> a(66)=809, a(90)=4165 >> a(87)=2062, ... . Here, the size of the terms roughly doubles over the interval of very few indices. The indices such that a(n[k]) >= 2*a(n[k-1]) are n[k] = 3, 5, 9, 17, 30, 46, 69, 89, ... .

This sequence first differs from A003278 at the 21st term, which is 92 here but 91 in A003278. Up to 91, each natural number n that did not appear in this sequence failed to do so because there were two smaller numbers n-a and n-2a, with A229037(n-a) and A229037(n-2a) both equal to 1. 91 is missing from this sequence; in other words, A229037(91) is not 1, because A229037(27) = 9 and A229037(59) = 5. - Jack W Grahl, Dec 28 2014

Charles R Greathouse IV and Chai Wah Wu, Table of n, a(n) for n = 1..434, First 266 terms from Charles R Greathouse IV.

Charles R Greathouse IV, [seqfan] Lexicographically first 3-free sequence (2014)

(Haskell)

a236246 n = a236246_list !! (n-1)

a236246_list = filter ((== 1) . a229037) [1..]

-- Reinhard Zumkeller, Apr 26 2014

(Python)

A236246_list, A229037_list = [], []

for n in range(10**6):

i, j, b = 1, 1, set()

while n-2*i >= 0:

b.add(2*A229037_list[n-i]-A229037_list[n-2*i])

i += 1

while j in b:

b.remove(j)

j += 1

A229037_list.append(j)

if j == 1:

A236246_list.append(n+1) # Chai Wah Wu, Dec 25 2014

Subsequence of A241673.

Sequence in context: A275482 A156799 A003278 * A004792 A167795 A322068

Adjacent sequences: A236243 A236244 A236245 * A236247 A236248 A236249

M. F. Hasler, Jan 20 2014

approved