cp's OEIS Frontend

A045576 Numbers k that divide 3^k + 2^k.

%I A045576 #42 Dec 01 2024 11:44:02
%S A045576 1,5,25,55,125,275,605,625,1375,3025,3125,6655,6875,15125,15625,30025,
%T A045576 31375,33275,34375,73205,75625,78125,150125,156875,166375,171875,
%U A045576 330275,345125,366025,378125,390625,439835,750625,784375,805255,831875
%N A045576 Numbers k that divide 3^k + 2^k.
%C A045576 For any j>=0, 5*A003598(j) is a term of the sequence. - _Benoit Cloitre_, Mar 08 2002
%C A045576 From _Robert Israel_, Jun 29 2017: (Start)
%C A045576 This is a semigroup: if m and n are in the sequence, then so is m*n.
%C A045576 If n is in the sequence and is divisible by prime p, then so is p*n.
%C A045576 The only prime powers in the sequence are the powers of 5.
%C A045576 Conjecture: Every member of the sequence except 1 is of the form p*m where p is prime and m is in the sequence. (End)
%C A045576 There are infinitely many primes p that divide some term in the sequence. Proof: Define the set A as all primes p such that a k where p divides 2^(5^k) + 3^(5^k) exists is finite. Since 2^(5^(k+1)) + 3^(5^(k+1)) is  2^(5^k) + 3^(5^k) multiplied by some positive integer a. It can be verified that gcd(a, 2^(5^k) + 3^(5^k)) is 5, so 2^(5^(k+1)) + 3^(5^(k+1)) has a larger number of different prime factors than 2^(5^k) + 3^(5^k). Therefore, A is infinite. For each prime q in A, suppose that q divides 2^(5^x) + 3^(5^x) for some x, then 5^x also divides it, so 5^x*q divides it as well, hence 5^x*q is a term of the sequence. The original theorem is proved. - _Yifan Xie_, Nov 14 2024
%H A045576 Robert Israel and Giovanni Resta, <a href="/A045576/b045576.txt">Table of n, a(n) for n = 1..1000</a> (first 205 terms from Robert Israel)
%p A045576 select(t -> 3 &^ t + 2 &^ t mod t = 0, [seq(i,i=1..10^6,2)]); # _Robert Israel_, Jun 29 2017
%o A045576 (PARI) isok(n) = ((3^n+2^n) % n) == 0; \\ _Michel Marcus_, Jun 29 2017
%o A045576 (PARI) isok(n)=(Mod(2,n)^n+Mod(3,n)^n)==0; \\ significantly more efficient
%o A045576 for(n=1,10^6,if(isok(n),print1(n,", "))); \\ _Joerg Arndt_, Aug 13 2017
%Y A045576 Cf. A003598, A007689.
%K A045576 nonn
%O A045576 1,2
%A A045576 _David W. Wilson_