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A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

%I A202274 #18 Dec 13 2024 11:08:40
%S A202274 1,2,4,8,16,25,32,64,128,256,512,1024,2048,4096,8192,16384,32768,
%T A202274 65536,131072,262144,524288,1048576,2097152,4194304,8388608,16777216,
%U A202274 33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,4294967296,8589934592,17179869184,34359738368,68719476736,137438953472
%N A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.
%C A202274 Original definition, now conjectural, is "Positive integers m in increasing order determined by these rules: a(1) = 1, for n>=1, if m is in the sequence then also are numbers h such that sigma(h) = 4m-1". This is certainly equal to the new definition if 25 is the only term that is not a power of 2, as there is no x such that sigma(x) = 99 = 4*25-1. - _Antti Karttunen_, Dec 13 2024
%C A202274 If 31 is only number h of form 2^k-1 for any k>=1 such that sigma(x) = h has solution for more than one value of x then a(n) is union number 25 with A000079 (powers of 2).
%C A202274 Numbers k such that A000203(k) is in A000225. If Goormaghtigh conjecture is valid, then it is certain that 25 is the only odd prime power (after 1) in this sequence. - _Antti Karttunen_, Dec 13 2024
%H A202274 Wikipedia, <a href="https://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>
%e A202274 These examples relate to the original definition:
%e A202274 m=1, 4m-1=3, sigma(h)=3 for h=2; number 2 is in sequence.
%e A202274 m=2, 4m-1=7, sigma(h)=7 for h=4; number 4 is in sequence.
%e A202274 m=4, 4m-1=15, sigma(h)=15 for h=8; number 8 is in sequence.
%e A202274 m=8, 4m-1=31, sigma(h)=31 for h=16 and 25; numbers 16 and 25 are in sequence.
%o A202274 (PARI) is_A202274(n) = ((x->!bitand(x,x+1))(sigma(n)));
%o A202274 for(n=1,2^20,if(is_A202274(n^2), print1(n^2,", ")); if(n>1 && is_A202274(2*((n-1)^2)), print1(2*((n-1)^2),", "))); \\ Remember to sort! - _Antti Karttunen_, Dec 13 2024
%Y A202274 Cf. A000079 (subsequence), A000203, A000225, A002191, A292369 (conjectured subsequence).
%Y A202274 Subsequence of A028982. Conjectured intersection of A028982 and A378983.
%Y A202274 Positions of 0's in A336694, A336695.
%Y A202274 Positions of 1's in A324294, A332459, A336692, A336693, A336696.
%Y A202274 Cf. also A202273.
%K A202274 nonn
%O A202274 1,2
%A A202274 _Jaroslav Krizek_, Dec 25 2011
%E A202274 Data section corrected (terms 1024, 2048 were duplicated), more terms added, and the name replaced with a new definition, with the original definition moved to the comments - _Antti Karttunen_, Dec 13 2024