A213257 a(1) = 1, a(2) = 2 and, for n > 2, a(n) is the smallest integer greater than a(n - 1) such that no three terms of the sequence form a geometric progression of the form {x, 2 x, 4 x}.

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 102, 103, 104, 105

Offset: 1

Author

John W. Layman, Jun 07 2012

Keywords

• nonn

Comments

Conjecture. The positive integers that are not in this sequence are given by the positions of 2 in the fixed-point of the morphism 0->01, 1->02, 2->03, 3->01 (see A191255). (This has been confirmed for over 5000 terms of A213257.) To illustrate, the fixed-point of the indicated morphism is {0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,1,0,1,0,2,0,...} and 2 occurs at positions {4,12,20,...}, integers that are missing in A213257.

The positive integers that are not in this sequence are listed in A213258.

For the sequence containing no 3-term arithmetic progression,see A003278.

Examples

			Given that the sequence begins {1, 2, 3, 5, 6, 7, 8, 9, 10, 11,...}, the next term, a(11), cannot be 12, because then the forbidden progression {3,6,12} would occur in the sequence.  13 is allowed, however, so a(11)=13.
		

Crossrefs

Cf. A003278, A191255, A213258.