A335120 The prime terms of A225563.

3, 5, 7, 11, 13, 17, 31, 41, 61, 97, 103, 137, 193, 241, 257, 409, 641, 769, 1021, 1361, 1543, 5441, 6529, 7681, 8161, 12289, 15361, 17477, 26113, 30841, 40961, 43691, 61441, 61681, 65537, 82241, 87041, 98689, 131071, 163841, 174761, 328961, 417793, 557057, 786433

Offset

1,1

Comments

Apparently, the prime terms of A225563 are relatively rare. For example, of the first 10^4 terms of A225563, only 23 are primes.

Alternatively, odd primes p such that phi(phi(p)), the number of primitive roots modulo p, is a power of two. Primes such that all odd prime divisors of p-1 are Fermat primes. - Paul Vanderveen, Mar 29 2022

From Jianing Song, May 29 2026: (Start)

Along with 2, primes p such that values taken by chi are all constructible numbers for all Dirichlet characters chi modulo p. General k satisfying this condition are listed in A396553.

Primes p such that p-1 is in A003401. In other words, p is either a Fermat prime or of the form d*2^n + 1, where d is in A045544, n >= 1. (End)

Links

Jianing Song, Table of n, a(n) for n = 1..539 (all terms < 10^1000)

Wikipedia, Constructible number.

Wikipedia, Dirichlet character.

Mathematica

totQ[n_] := PrimeQ[n] && Module[{it = Most@FixedPointList[EulerPhi, n], sum, x}, sum = Plus @@ it; If[OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, it}], x][[1 +sum/2]] > 0]]; Select[Range[10^3], totQ]

Prog

(PARI) ispower2(n) = (bitand(n, n-1) == 0)
isA335120(p) = (p>2 && isprime(p) && ispower2(eulerphi(p-1))) \\ Jianing Song, May 29 2026

Crossrefs

Cf. A225563, A003401, A045544, A396553, A396555 (complement in primes).

Sequence in context: A228118 A136186 A023210 * A153601 A153602 A136187

Adjacent sequences: A335117 A335118 A335119 * A335121 A335122 A335123

Keyword

nonn

Author

Amiram Eldar, May 24 2020

Status

approved