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

%I A235264 #35 Dec 23 2024 14:53:43
%S A235264 1,3,5,7,9,15,17,21,31,33,51,63,65,73,85,127,129,195,255,257,273,341,
%T A235264 455,511,513,585,771,819,1023,1025,1057,1285,1365,2047,2049,3075,3591,
%U A235264 3855,4095,4097,4161,4369,4681,5461
%N A235264 Tileable numbers: base-2 representation, considered as a fixed disconnected polyomino, tiles all places >= 0.
%H A235264 Charlie Neder, <a href="/A235264/b235264.txt">Table of n, a(n) for n = 1..2314</a> (First 701 terms from David W. Wilson)
%H A235264 Charlie Neder, <a href="/A235264/a235264.txt">Proof of characterization of this sequence</a>
%H A235264 Allan Wechsler, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-November/011891.html">A possible characterization of A125121</a> (Original idea)
%F A235264 Numbers n such that 2-adic m = -1/n exists and 2-adic product m*n involves no carries.
%F A235264 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.
%F A235264 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.
%F A235264 First conjecture is equivalent to the 2-adic definition. - _Charlie Neder_, Nov 04 2018
%F A235264 Second conjecture is true, see Neder link. - _Charlie Neder_, Dec 04 2018
%e A235264 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.
%Y A235264 Conjecturally, subset of A006995 (base-2 palindromes).
%K A235264 nonn,nice
%O A235264 1,2
%A A235264 _David W. Wilson_, Jan 05 2014