cp's OEIS Frontend

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

%I A370452 #27 Jun 25 2024 05:24:20
%S A370452 3,5,7,9,13,17,25,31,37,49,53,61,73,81,97,121,127,157,193,241,277,313,
%T A370452 337,361,397,421,457,541,577,613,625,661,673,733,757,841,877,997,1093,
%U A370452 1153,1201,1213,1237,1249,1321,1381,1453,1621,1657,1681,1753,1873,1933,1993,2017,2137,2341,2401,2473,2557,2593,2797,2857
%N A370452 Prime powers of the form 2*p^k-1, where p is prime and k >= 1.
%C A370452 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
%C A370452 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.
%H A370452 Keith J. Bauer, <a href="/A370452/a370452.txt">The monoid (F_n[j], *)/(a + bj ~ b + aj) is isomorphic to (F_n X F_(n+1)/2, *), when such fields exist</a>.
%e A370452 3 = 2*2^1 - 1 = 3^1;
%e A370452 5 = 2*3^1 - 1 = 5^1;
%e A370452 7 = 2*2^2 - 1 = 7^1;
%e A370452 9 = 2*5^1 - 1 = 3^2.
%p A370452 filter:= n -> nops(numtheory:-factorset(n))=1 and nops(numtheory:-factorset((n+1)/2))=1:
%p A370452 select(filter, [seq(i,i=3..10000,2)]); # _Robert Israel_, Feb 20 2024
%t A370452 Select[Range[3000], PrimePowerQ[#] && PrimePowerQ[(# + 1)/2] &] (* _Amiram Eldar_, Feb 19 2024 *)
%o A370452 (PARI) isok(q) = isprimepower(q) && (q%2) && isprimepower((q+1)/2); \\ _Michel Marcus_, Jun 14 2024
%Y A370452 Cf. A178490, A246655 (prime powers).
%K A370452 easy,nonn
%O A370452 1,1
%A A370452 _Keith J. Bauer_, Feb 18 2024