A205506 - OEIS

A205506

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

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, 28, 372, 14, 47, 25, 569, 646, 141, 129, 278, 5, 421, 224, 629, 26, 424, 1081, 688, 246, 736, 4392, 124, 484, 759, 791, 4401, 863, 2854, 410, 1044, 22, 848, 1402, 2006

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OFFSET

1,1

COMMENTS

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.

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

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

LINKS

Table of n, a(n) for n=1..59.

FORMULA

a(n) = A085398(6*A003586(n)). - Jinyuan Wang, Jan 01 2023

a(n) is smallest positive m such that Phi(A033845(n),m) is prime. - Chai Wah Wu, Sep 16 2024

EXAMPLE

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

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

...

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

MATHEMATICA

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

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

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

PROG

(Python)

from itertools import count

from sympy import isprime, integer_log

def A205506(n):

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

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

while kmax-kmin > 1:

kmid = kmax+kmin>>1

if f(kmid) <= kmid:

kmax = kmid

else:

kmin = kmid

return kmax

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

k = bisection(f, n, n)

return next(filter(lambda m:isprime(1-m**k+m**(k<<1)), count(2))) # Chai Wah Wu, Oct 22 2024

CROSSREFS

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

Sequence in context: A134339 A162299 A281552 * A110141 A339489 A293443

Adjacent sequences: A205503 A205504 A205505 * A205507 A205508 A205509

KEYWORD

nonn,hard

AUTHOR

Lei Zhou, Feb 01 2012

STATUS

approved