cp's OEIS Frontend

A343716 Numbers k such that k^2 divides 5^k - 4^k - 3^k.

%I A343716 #9 May 16 2021 21:28:40
%S A343716 1,2,4,6,8,10,12,18,24,26,30,36,50,72,74,90,130,150,338,370,450,650,
%T A343716 962,1402,1850,2738,2974,4810,8450,12506,24050,35594,64382,68450,
%U A343716 184706,270150,312650,462722,889850,11568050
%N A343716 Numbers k such that k^2 divides 5^k - 4^k - 3^k.
%C A343716 If it exists, a(41) > 10^9.
%C A343716 Of the 4*A015616(10) = 4*109 = 436 integer sequences of one of the forms
%C A343716      Numbers k such that k^2 | A^k + B^k + C^k,
%C A343716      Numbers k such that k^2 | A^k + B^k - C^k,
%C A343716      Numbers k such that k^2 | A^k - B^k + C^k,
%C A343716   or Numbers k such that k^2 | A^k - B^k - C^k
%C A343716 such that 0 < C < B < A <= 10 and gcd(A,B,C)=1, this one appears to have the largest number of terms.
%C A343716 By comparison, A127074 (k such that k^2 | 3^k - 2^k - 1) and A343115 (k such that k^2 | 5^k - 3^k - 2^k) seem unlikely to have any terms beyond A127074(9)=17807 and A343115(14)=876, respectively. Only 25 of the 436 above sequences have any 4-, 5-, or 6-digit terms at all.
%C A343716 a(41) > 10^11 if it exists. - _Chai Wah Wu_, May 16 2021
%e A343716 5^2 - 4^2 - 3^2 = 25 - 16 - 9 = 0, which is divisible by 2^2 = 4, so 2 is a term.
%e A343716 5^18 - 4^18 - 3^18 = 3745590368400 = 11560464100 * 18^2, so 18 is a term.
%o A343716 (Python)
%o A343716 def afind(startat=1, limit=10**9):
%o A343716   for k in range(startat, limit+1):
%o A343716     kk = k*k
%o A343716     if (pow(5, k, kk) - pow(4, k, kk) - pow(3, k, kk))%kk == 0:
%o A343716       print(k, end=", ")
%o A343716 afind(limit=10**5) # _Michael S. Branicky_, May 16 2021
%Y A343716 Cf. A015616, A127074, A343115.
%K A343716 nonn,more
%O A343716 1,2
%A A343716 _Jon E. Schoenfield_, May 08 2021