A045544 Odd values of n for which a regular n-gon can be constructed by compass and straightedge.

3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

Offset: 1

Author

Ken Takusagawa

Keywords

• hard
• nonn
• nice

Comments

If there are no more Fermat primes, then 4294967295 is the last term in the sequence.

From Daniel Forgues, Jun 17 2011: (Start)

The 31 = 2^5 - 1 terms of this sequence are the nonempty products of distinct Fermat primes. The 5 known Fermat primes are in A019434.

Prepending the empty product, i.e., 1, to this sequence gives A004729.

The initial term for this sequence is thus a(1) (offset=1), since a(0) should correspond to the omitted empty product, term a(0) of A004729.

Rows 1 to 31 of Sierpiński's triangle, if interpreted as a binary number converted to decimal (A001317), give a(1) to a(31). (End)

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 73 at pp. 181-182.

Links

• Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m.
• OEIS Wiki, Constructible odd-sided polygons.
• OEIS Wiki, Sierpinski's triangle.

Crossrefs

Cf. A019434. Essentially same as A004729.

Coincides with A001317 for the first 31 terms only. - Robert G. Wilson v, Dec 22 2001

Cf. A053576.

Programs

Mathematica

Union[Times@@@Rest[Subsets[{3,5,17,257,65537}]]] (* Harvey P. Dale, Sep 20 2011 *)

Formula

Each term is the product of distinct odd Fermat primes.

Sum_{n>=1} 1/a(n) = -1 + Product_{n>=1} (1+1/A019434(n)) = 0.7007354948... >= 1003212011/1431655765 = sigma(2^32-1)/(2^32-1) - 1, with equality if there are only five Fermat primes (A019434). - Amiram Eldar, Jan 22 2022