A370452 Prime powers of the form 2*p^k-1, where p is prime and k >= 1.

3, 5, 7, 9, 13, 17, 25, 31, 37, 49, 53, 61, 73, 81, 97, 121, 127, 157, 193, 241, 277, 313, 337, 361, 397, 421, 457, 541, 577, 613, 625, 661, 673, 733, 757, 841, 877, 997, 1093, 1153, 1201, 1213, 1237, 1249, 1321, 1381, 1453, 1621, 1657, 1681, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2401, 2473, 2557, 2593, 2797, 2857

Offset: 1

Author

Keith J. Bauer, Feb 18 2024

Keywords

• easy
• nonn

Comments

Also, sizes of finite fields such that halving the size of their unit group is also the unit group of a field. - Keith J. Bauer, Jun 20 2024

Original motivation for this sequence: let k be a term of this sequence. Then consider the finite field of k elements, denoted by F_k. Adjoin the hyperbolic unit j^2 = 1 to F_k to form a ring whose elements are of the form a + bj for a, b in F_k. Let M be the multiplication monoid of F_k[j] and let ~ be the equivalence relation on the elements of M defined by a + bj ~ b + aj (with no further unnecessary equivalences). Then M/~ is isomorphic to the multiplication monoid of the ring F_k x F_(k+1)/2 and therefore there exists a ring with M/~ as its multiplication. For prime powers k not in this sequence, no such ring will exist. See the link for a proof of this fact.

Examples

			3 = 2*2^1 - 1 = 3^1;
5 = 2*3^1 - 1 = 5^1;
7 = 2*2^2 - 1 = 7^1;
9 = 2*5^1 - 1 = 3^2.
		

Links

• Keith J. Bauer, The monoid (F_n[j], *)/(a + bj ~ b + aj) is isomorphic to (F_n X F_(n+1)/2, *), when such fields exist.

Crossrefs

Cf. A178490, A246655 (prime powers).

Programs

Maple

filter:= n -> nops(numtheory:-factorset(n))=1 and nops(numtheory:-factorset((n+1)/2))=1:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Feb 20 2024

Mathematica

Select[Range[3000], PrimePowerQ[#] && PrimePowerQ[(# + 1)/2] &] (* Amiram Eldar, Feb 19 2024 *)

PARI

isok(q) = isprimepower(q) && (q%2) && isprimepower((q+1)/2); \\ Michel Marcus, Jun 14 2024