A320481 Primes in A301916 but not in A045318.

2, 769, 1297, 6529, 7057, 8017, 8737, 12097, 12289, 13297, 13441, 14929, 15073, 15361, 15937, 16273, 18913, 19441, 20593, 21601, 21649, 22273, 22369, 23857, 25633, 26017, 26449, 26497, 27793, 28513, 30529, 31249, 34369, 34849, 36913, 37057, 37441, 37633, 38833, 38977, 39409

Offset: 1

Author

N. J. A. Sloane, Oct 17 2018

Keywords

• nonn

Comments

Is there a simpler characterization of these primes?

Answer from Don Reble, Oct 25 2018: (Start)

Let POT(x) be the largest power of 2 which divides x (A006519).

Apart from the initial 2, this sequence consists of those primes P such that

2 <= POT(the order of 3 modulo P) <= POT(P-1)/8.

The condition "2 <=" ensures that P divides some 3^k+1, and the condition "<= POT(P-1)/8" is so that 3 has an eighth root modulo P. A062117 is the order of 3 modulo prime(n). (End)

Comments from Richard Bumby, Nov 12 2018: (Start)

When considering methods for finding square roots mod p one is led to filtering the nonzero elements by the power of 2 dividing the multiplicative order of the element. The lowest level -- elements of odd order -- have easily computed square roots, and the square roots of other elements can be found if you can discover at least one element at a higher level.

To say that "x^8 = 3 has no solution mod p" is to say that 3 is in one of the top three levels and that there are more than 3 levels (so that 8 divides p-1).

To say that primes "divide numbers of the form 3^k + 1" is to say that -1 is a power of 3 mod p, or that 3 is not at the lowest level. If there are only four levels (9 mod 16), these statements are equivalent. Otherwise, the two statements are different. An interesting case has 3 at the second level, so that (-3) has odd order allowing cube roots of unity to be found quickly.

I was told that Odoni had some results on finding the number of primes with k levels for which a given number (e.g., 3) is at level j, but I never tracked down a reference. If the asymptotic behavior is what one would expect, A045318 and A301916 are really far from being "almost the same", except in the trivial sense of "zero density". (End)

References

• Georg Fischer, email to N. J. A. Sloane, Oct 16 2018.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006519, A045317, A045318, A062117, A301916, A301917.

Programs

Mathematica

Select[Prime@Range@4200,PowerModList[3,1/8,#]!={}&&IntegerQ@MultiplicativeOrder[3,#,-1]&] (* Giorgos Kalogeropoulos, Feb 23 2022 *)

Extensions

More terms from Michel Marcus, Oct 17 2018