A236246 Indices n for which A229037(n)=1.

1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41, 82, 83, 85, 86, 92, 93, 96, 105, 111, 112, 115, 116, 122, 177, 236, 237, 244, 245, 247, 266, 267, 270, 276, 277, 283, 294, 301, 302, 347, 558, 628, 638, 646, 647, 649, 655, 669, 674, 685, 686

Offset: 1

Author

M. F. Hasler, Jan 20 2014

Keywords

• nonn

Comments

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

Links

• 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)

Crossrefs

Subsequence of A241673.

Programs

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