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

1, 4, 8, 9, 25, 27, 32, 36, 121, 125, 128, 2704, 5041, 5184

Offset: 1

Author

Gonzalo Martínez, Mar 23 2024

Keywords

• nonn
• more

Comments

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.

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).

On the other hand, 4, 8, 32 and 128 are terms of A000079.

The representation for each term is as follows.

1 = 1! + 0

4 = 1! + 3 = 2! + 2

8 = 3! + 2

9 = 1! + 8 = 3! + 3

25 = 4! + 1

27 = 3! + 21 = 4! + 3

32 = 4! + 8

36 = 2! + 34

121 = 5! + 1

125 = 5! + 5

128 = 5! + 8

2704 = 5! + 2584

5041 = 7! + 1

5184 = 7! + 144

Examples

			128 is a term because 128 = 2^7 and 128 = 5! + 8, where 8 is a Fibonacci number.
		

Crossrefs

Cf. A000045, A000142, A001597, A227644, A272575.