A383649 Numbers k such that A206369(k) is prime.

3, 4, 6, 8, 9, 16, 18, 49, 64, 81, 98, 162, 169, 338, 625, 729, 1024, 1250, 1458, 4096, 4489, 6241, 8978, 12482, 14641, 19321, 22801, 26569, 29282, 37249, 38642, 45602, 53138, 65536, 74498, 113569, 121801, 143641, 208849, 227138, 243602, 262144, 287282, 292681, 375769, 413449, 417698

Offset: 1

Author

Shreyansh Jaiswal, May 04 2025

Keywords

• nonn

Comments

The corresponding primes are 2, 3, 2, 5, 7, 11, 7, 43, 43, 61, 43, 61, 157, 157, 521... Sorting and removing duplicates from this sequence of primes appears to give A127727.

All the terms are either prime powers or twice prime powers since A206369 is multiplicative and A206369(n) = 1 only for n = 1 and 2. 3 is the only prime term since A206369(p) = p-1 for a prime p. - Amiram Eldar, May 04 2025

Examples

			3 is a term since A206369(3) = 2, and 2 is a prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..122 from Shreyansh Jaiswal)

Crossrefs

Cf. A127727, A206369.

Programs

Mathematica

seq[max_] := Module[{s = {3, 6}, e}, Do[e = Select[Range[2, Floor[Log[p, max]]], PrimeQ[Sum[(-1)^(# - k)*p^k, {k, 0, #}]] &]; s = Join[s, p^e]; If[p > 2, e = Select[e, # <= Floor[Log[p, max/2]] &]; s = Join[s, 2*p^e]], {p, Prime[Range[Floor[Sqrt[max]]]]}]; Sort[s]]; seq[500000] (* Amiram Eldar, May 04 2025 *)

PARI

isok(k) = ispseudoprime(sumdiv(k, d, eulerphi(k/d) * issquare(d))); \\ Michel Marcus, May 04 2025

Python

from math import prod; from sympy import *
def ok(n): return isprime(prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1), p+1)) for p, e in factorint(n).items())) # using code from Chai Wah Wu in A206369
print([k for k in range(1,10**5) if ok(k)])