A235264 Tileable numbers: base-2 representation, considered as a fixed disconnected polyomino, tiles all places >= 0.

1, 3, 5, 7, 9, 15, 17, 21, 31, 33, 51, 63, 65, 73, 85, 127, 129, 195, 255, 257, 273, 341, 455, 511, 513, 585, 771, 819, 1023, 1025, 1057, 1285, 1365, 2047, 2049, 3075, 3591, 3855, 4095, 4097, 4161, 4369, 4681, 5461

Offset: 1

Author

David W. Wilson, Jan 05 2014

Keywords

• nonn
• nice

Examples

			n = 3855 has 2-adic representation .10100000101, and negative reciprocal repeating 2-adic m = .(1100110000000000)...  The 2-adic product n*m = -1 = .(1)... involves no carries, so n is tileable.
		

Links

• Charlie Neder, Table of n, a(n) for n = 1..2314 (First 701 terms from David W. Wilson)
• Charlie Neder, Proof of characterization of this sequence
• Allan Wechsler, A possible characterization of A125121 (Original idea)

Crossrefs

Conjecturally, subset of A006995 (base-2 palindromes).

Formula

Numbers n such that 2-adic m = -1/n exists and 2-adic product m*n involves no carries.

Conjecturally, a(n) = (2^k-1)/m where k, m >= 1, and base-2 product m*a(n) involves no carries. Confirmed for a(n) <= 2^20.

Conjecturally, a(n) is of the form Product (2^(d_i*b_i)-1)/(2^b_i-1) where d_i >= 1, b_i >= 2, and d_i*b_i | d_(i+1). Confirmed for a(n) <= 2^20.

First conjecture is equivalent to the 2-adic definition. - Charlie Neder, Nov 04 2018

Second conjecture is true, see Neder link. - Charlie Neder, Dec 04 2018