A085692 -id:A085692 - cp's OEIS frontend

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

Gonzalo Martínez, Mar 23 2024

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! + 0
= 1! + 3 = 2! + 2
= 3! + 2
= 1! + 8 = 3! + 3
= 4! + 1
= 3! + 21 = 4! + 3
= 4! + 8
= 2! + 34
= 5! + 1
= 5! + 5
= 5! + 8
= 5! + 2584
= 7! + 1
= 7! + 144

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

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

A373970 Squares equal to the sum of a cube and a factorial number.

Original entry on oeis.org

1, 9, 25, 121, 784, 1024, 5041, 363609, 542939080336, 160351569000000, 51312167754301440000, 65249379457597440000, 449519240612413440000, 957230928468541440000, 4797276365676433637376, 16367773504928806600704, 308090303428827709440000

Offset: 1

Martin Ehrenstein, Jun 24 2024

			1 = 0^3 + 1!.
25 = 1^3 + 4!.
784 = 4^3 + 6!.
5041 = 1^3 + 7!.

Cf. A000142, A000290, A000578, A373218.

A085692 is a subsequence.

is1(n) = {my(sq = n^2, f = 1, k = 1); while(f <= sq && !ispower(sq - f, 3), k++; f *= k); f <= sq;}
lista(kmax) = for(k = 1, kmax, if(is1(k), print1(k^2, ", "))); \\ Amiram Eldar, Jun 24 2024

a(10) from Amiram Eldar, Jun 24 2024
a(11)-a(16) from David A. Corneth, Jun 24 2024
a(17) from Michael S. Branicky, Jun 26 2024

cp's OEIS Frontend

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

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A373970 Squares equal to the sum of a cube and a factorial number.

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