cp's OEIS Frontend

A371223 Perfect powers (A001597) equal to the sum of a factorial number (A000142) and a Fibonacci number (A000045).

%I A371223 #31 Mar 24 2024 12:30:36
%S A371223 1,4,8,9,25,27,32,36,121,125,128,2704,5041,5184
%N A371223 Perfect powers (A001597) equal to the sum of a factorial number (A000142) and a Fibonacci number (A000045).
%C A371223 Listed terms are 1, 2^2, 2^3, 3^2, 5^2, 3^3, 2^5, 6^2, 11^2, 5^3, 2^7, 52^2, 71^2 and 72^2.
%C A371223 It is observed that 4, 8, 25, 121 and 5041 are also terms of A227644 (Perfect powers equal to the sum of two factorial numbers), where in turn 25, 121 and 5041 are terms of A085692 (Brocard's problem), while the first 4 terms and 36 are part of A272575 (Perfect powers that are the sum of two Fibonacci numbers).
%C A371223 On the other hand, 4, 8, 32 and 128 are terms of A000079.
%C A371223 The representation for each term is as follows.
%C A371223      1 = 1! + 0
%C A371223      4 = 1! + 3 = 2! + 2
%C A371223      8 = 3! + 2
%C A371223      9 = 1! + 8 = 3! + 3
%C A371223     25 = 4! + 1
%C A371223     27 = 3! + 21 = 4! + 3
%C A371223     32 = 4! + 8
%C A371223     36 = 2! + 34
%C A371223    121 = 5! + 1
%C A371223    125 = 5! + 5
%C A371223    128 = 5! + 8
%C A371223   2704 = 5! + 2584
%C A371223   5041 = 7! + 1
%C A371223   5184 = 7! + 144
%e A371223 128 is a term because 128 = 2^7 and 128 = 5! + 8, where 8 is a Fibonacci number.
%Y A371223 Cf. A000045, A000142, A001597, A227644, A272575.
%K A371223 nonn,more
%O A371223 1,2
%A A371223 _Gonzalo Martínez_, Mar 23 2024