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%I #31 Mar 22 2024 06:48:25
%S 10,14,22,38,44,92,110,130,134,136,152,170,184,248,250,262,284,376,
%T 410,442,632,730,752,884,988,1012,1052,1276,1292,1370,1628,2144,2168,
%U 2272,2332,2528,3068,4064,4124,5210,6112,6364,6556,7372,8198,8312,8384,8648
%N Even k such that k - A001065(k) = 2^m (for some m > 0).
%C Tianxin Cai conjectured that the sequence is infinite.
%C When p is prime, A001065(2p) = 1 + 2 + p = 3 + p. 2p - A001065(2p) = 2^m iff 2^m + 3 = p. Therefore if A057733 is infinite, Cai's conjecture is correct.
%C In general, for j = 2, 3, ..., if the number of primes of the form 2^m + 2^j - 1 is infinite, then Cai's conjecture is correct.
%C When 2^p - 1 is prime, let k = 2^p*(2^p - 1). A001065(k) = 1 + 2 + 2^2 + ... + 2^p + 2^p - 1 + 2(2^p - 1) + 2^2*(2^p - 1) + ... + 2^(p - 1)*(2^p - 1) = 2^(p + 1) - 1 + (2^p - 1)^2 = 2^(2p). k - A001065(k) = -2^p. Therefore if the number of Mersenne primes (A000668) is infinite, then there are infinitely many even k such that k - A001065(k) = -2^p.
%H Amiram Eldar, <a href="/A286843/b286843.txt">Table of n, a(n) for n = 1..893</a> (terms below 10^11)
%e 10 is a term as 10 - A001065(10) = 10 - 8 = 2.
%e 22 is a term as 22 - A001065(22) = 22 - 14 = 2^3.
%t Select[Table[2n, {n, 1, 5000}], DivisorSigma[1, 2# - DivisorSigma[1, #]] + 1 == 2(2# - DivisorSigma[1, #]) > 2 &] (* or *)
%t Select[2 Range[5000], IntegerQ@ Log2[2 # - DivisorSigma[1,#]] && !IntegerQ@ Log2@ # &] (* _Giovanni Resta_, Aug 07 2017 *)
%o (PARI) ispower2(n) = n > 1 && n >> valuation(n, 2) == 1;
%o is(n) = !(n%2) && ispower2(2*n - sigma(n)); \\ _Amiram Eldar_, Mar 22 2024
%Y Cf. A000079, A001065, A005843, A057733.
%K nonn
%O 1,1
%A _XU Pingya_, Aug 01 2017
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