[seqfan] A086341

[Uwe Lauth]

Hello,

I suggest to add the following comment to A086341:
"With the exception of 1,
these are the integers whose squares are Proth numbers."

Let me briefly describe this.

Any odd integer can be written as n = k * 2^m + 1, with odd k.
Proth numbers (A080075) are those where k < 2^m.

Some Proth numbers are also square numbers:
9, 25, 49, 81, 225, 289, 961, ...
I suggest to call them Proth squares.

That sequence is not in the OEIS. But the sequence of their roots,
3, 5, 7, 9, 15, 17, 31, ...
is A086341 where the first two numbers (1,3) are omitted.

A086341 is the powers of 2, plus/minus 1.
Their squares (with the exception of 1) are Proth numbers:
(2^(m-1) +- 1)^2 = k * 2^m + 1  where  k = 2^(m-2) +- 1

No other square can be a Proth number. To show that,
write the root b = a * 2^(m-1) +- 1 (with m>1 and odd a).
b^2 = k * 2^m + 1  where  k = a^2 * 2^(m-2) +- a
k becomes "too big" already for a = 3, since 9 > 2^2.

(Note: that proof does not work for the first few squares,
but these can be verified by hand.)

I verified this numerically up to b = 2^32-1.

Another conclusion:
The set of Proth squares is infinite.

Kind regards,
Uwe