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

1, 2, 4, 8, 16, 25, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472

Offset: 1

Author

Jaroslav Krizek, Dec 25 2011

Keywords

• nonn

Comments

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

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).

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

Examples

			These examples relate to the original definition:
m=1, 4m-1=3, sigma(h)=3 for h=2; number 2 is in sequence.
m=2, 4m-1=7, sigma(h)=7 for h=4; number 4 is in sequence.
m=4, 4m-1=15, sigma(h)=15 for h=8; number 8 is in sequence.
m=8, 4m-1=31, sigma(h)=31 for h=16 and 25; numbers 16 and 25 are in sequence.
		

Links

• Wikipedia, Goormaghtigh conjecture

Crossrefs

Cf. A000079 (subsequence), A000203, A000225, A002191, A292369 (conjectured subsequence).

Subsequence of A028982. Conjectured intersection of A028982 and A378983.

Positions of 0's in A336694, A336695.

Positions of 1's in A324294, A332459, A336692, A336693, A336696.

Cf. also A202273.

Programs

PARI

is_A202274(n) = ((x->!bitand(x,x+1))(sigma(n)));
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

Extensions

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