A205506 - OEIS

%I #37 Oct 23 2024 00:43:18

%S 2,2,6,2,3,5,7,3,4,3,6,93,2,88,5,33,5,196,15,106,174,196,14,342,207,

%T 28,372,14,47,25,569,646,141,129,278,5,421,224,629,26,424,1081,688,

%U 246,736,4392,124,484,759,791,4401,863,2854,410,1044,22,848,1402,2006

%N Least positive integer m > 1 such that 1 - m^k + m^(2*k) is prime, where k=A003586(n).

%C 1 - m^k + m^(2*k) equals Phi(6*k,m) when k=2^p*3^q, p>=0, q>=0, which may be prime numbers for certain positive integer m>1.

%C The Mathematica program given here generates the first 33 terms.  Further terms were generated by OpenPFGW.

%C a(62)=7426, while A003586(62)=3^8=6561.

%F a(n) = A085398(6*A003586(n)). - _Jinyuan Wang_, Jan 01 2023

%F a(n) is smallest positive m such that Phi(A033845(n),m) is prime. - _Chai Wah Wu_, Sep 16 2024

%e n=1, A003586(1)=1, when m=2, 1-2^1+2^2=3 is prime, so a(1)=2;

%e n=2, A003586(2)=2, when m=2, 1-2^2+2^4=13 is prime, so a(2)=2;

%e ...

%e n=7, A003586(7)=9, when m=7, 1-7^9+7^18=1628413557556843 is prime, so a(7)=7.

%t fQ[n_] := n == 3 EulerPhi@n; a = Select[6 Range@500, fQ]/6; l =

%t Length[a]; Table[m = a[[j]]; i = 1;

%t While[i++; cp = 1 - i^m + i^(2*m); ! PrimeQ[cp]]; i, {j, 1, l}]

%o (Python)

%o from itertools import count

%o from sympy import isprime, integer_log

%o def A205506(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))

%o     k = bisection(f,n,n)

%o     return next(filter(lambda m:isprime(1-m**k+m**(k<<1)),count(2))) # _Chai Wah Wu_, Oct 22 2024

%Y Cf. A003586, A033845, A056993, A085398, A153438.

%K nonn,hard

%O 1,1

%A _Lei Zhou_, Feb 01 2012