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%I #9 Nov 25 2019 00:57:53
%S 1,11,45,181,362,724,1448,2896,11585,23170,741455,1482910,11863283,
%T 23726566,189812531,379625062,97184015999,194368031998,3109888511975,
%U 99516432383215,199032864766430,1592262918131443,3184525836262886
%N Integers n such that n^2 + k is a Mersenne number 2^m - 1 for some k < n and m odd.
%C Subset of A017910.
%C The corresponding Mersenne number exponents are given by A144931.
%C From _Gil Broussard_, Sep 12 2009: (Start)
%C It appears that the terms of this sequence are the only numbers with this property: the binary expansion of a(n) is identical to the first ceiling(log_2(a(n))) nonzero digits of the binary expansion of 1/a(n). In other words, if the binary expansion of a(n) is 6 digits, then the first 6 nonzero digits of the binary expansion of 1/a(n) is identical for some a(n).
%C For example:
%C a(2)=11=binary 1011 which is 4 digits long and equivalent to the first 4 digits of its binary reciprocal (after the initial zeros):
%C 1/a(2) = binary .000[1011]101000101110100010111010...
%C Table of a(2) to a(11):
%C 11 1011 -> .000[1011]1010001011101000101110100010111010001011...
%C 45 101101 -> .00000[101101]100000101101100000101101100000101101...
%C 181 10110101 -> .0000000[10110101]00001001111001101000101010011011...
%C 362 101101010 -> .00000000[101101010]000100111100110100010101001101...
%C 724 1011010100 -> .000000000[1011010100]0010011110011010001010100110...
%C 1448 10110101000 -> .0000000000[10110101000]01001111001101000101010011...
%C 2896 101101010000 -> .00000000000[101101010000]100111100110100010101001...
%C 11585 10110101000001 -> .0000000000000[10110101000001]01111001100110100100...
%C 23170 101101010000010 -> .00000000000000[101101010000010]111100110011010010...
%C 741455 10110101000001001111 -> .0000000000000000000[10110101000001001111]01100110...
%C (End)
%o (PARI) forstep(m=1,10^6,2,n=sqrtint(2^m-1);if(2^m-1-n^2<n,print1(n,", ")))
%Y Cf. A000225, A017910, A144931.
%K nonn
%O 1,2
%A _Reikku Kulon_, Sep 25 2008
%E Edited by _Max Alekseyev_, Oct 12 2009
%E Edited by _Charles R Greathouse IV_, Mar 23 2010
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