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

%I A383649 #22 May 09 2025 22:25:14
%S A383649 3,4,6,8,9,16,18,49,64,81,98,162,169,338,625,729,1024,1250,1458,4096,
%T A383649 4489,6241,8978,12482,14641,19321,22801,26569,29282,37249,38642,45602,
%U A383649 53138,65536,74498,113569,121801,143641,208849,227138,243602,262144,287282,292681,375769,413449,417698
%N A383649 Numbers k such that A206369(k) is prime.
%C A383649 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.
%C A383649 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
%H A383649 Amiram Eldar, <a href="/A383649/b383649.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..122 from Shreyansh Jaiswal)
%e A383649 3 is a term since A206369(3) = 2, and 2 is a prime.
%t A383649 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 *)
%o A383649 (Python)
%o A383649 from math import prod; from sympy import *
%o A383649 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
%o A383649 print([k for k in range(1,10**5) if ok(k)])
%o A383649 (PARI) isok(k) = ispseudoprime(sumdiv(k, d, eulerphi(k/d) * issquare(d))); \\ _Michel Marcus_, May 04 2025
%Y A383649 Cf. A127727, A206369.
%K A383649 nonn
%O A383649 1,1
%A A383649 _Shreyansh Jaiswal_, May 04 2025