A227644 -id:A227644 - cp's OEIS frontend

A114377 Perfect powers equal to the sum of three factorial numbers.

4, 8, 9, 27, 32, 36, 49, 128, 841, 5184

Offset: 1

Giovanni Teofilatto, Feb 10 2006

nonn

a(11), if it exists, is larger than 10^100.

a(11), if it exists, is larger than 1024!. - Filip Zaludek, Sep 08 2018

			a(1) = 4 because 4 = 1! + 1! + 2!;
a(2) = 8 because 8 = 1! + 1! + 3!;
a(3) = 9 because 9 = 1! + 2! + 3!.

Giovanni Resta, Decompositions for a(1)-a(10)

Cf. A082875, A227644-A227651,

Offset corrected by Arkadiusz Wesolowski, Mar 06 2013

Example corrected by and a(9)-a(10) from Giovanni Resta, Jul 19 2013

A227651 Perfect powers equal to the sum of 10 factorial numbers.

Original entry on oeis.org

16, 25, 27, 32, 36, 49, 64, 81, 100, 125, 128, 144, 169, 196, 216, 225, 256, 289, 324, 343, 400, 441, 484, 512, 529, 625, 676, 729, 784, 841, 900, 1000, 1024, 1089, 1156, 1225, 1331, 1444, 1521, 1600, 1681, 1728, 1936, 2048, 2187, 2197, 2304, 2401, 2916, 3025

Offset: 1

Giovanni Resta, Jul 19 2013

nonn

a(240), if it exists, is larger than 10^50.

			25411681 = 71^4 = 1! + 7! + 7! + 10! + 10! + 10! + 10! + 10! + 10! + 10!.

Giovanni Resta, Table of n, a(n) for n = 1..239 (terms < 10^50)
Giovanni Resta, Decompositions for a(1)-a(239)

Cf. A114377, A227644-A227650.

A227650 Perfect powers equal to the sum of 9 factorial numbers.

Original entry on oeis.org

9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 400, 441, 512, 729, 784, 900, 961, 1000, 1024, 1089, 1156, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 2048, 2187, 2197, 2209, 2304, 2916, 3125, 3844, 4356

Offset: 1

Giovanni Resta, Jul 19 2013

nonn

a(178), if it exists, is larger than 10^100.

			1815848 = 122^3 = 2! + 3! + 6! + 6! + 9! + 9! + 9! + 9! + 9!.

Giovanni Resta, Table of n, a(n) for n = 1..177 (terms < 10^100)
Giovanni Resta, Decompositions for a(1)-a(177)

Cf. A114377, A227644-A227651.

A227645 Perfect powers equal to the sum of 4 factorial numbers.

Original entry on oeis.org

4, 8, 9, 16, 27, 32, 125, 128, 169, 243, 361, 729, 961, 1444, 10201, 403225, 725904

Offset: 1

Giovanni Resta, Jul 19 2013

nonn

a(18), if it exists, is larger than 10^100.

			729 = 3^6 = 1! + 2! + 3! + 6!.

Giovanni Resta, Decompositions of a(1)-a(17)

Cf. A114377, A227644-A227651, A082920.

A227646 Perfect powers equal to the sum of 5 factorial numbers.

Original entry on oeis.org

8, 9, 16, 25, 32, 36, 125, 128, 144, 216, 243, 289, 729, 1444, 1681, 2304, 3600, 5832, 45369, 121104, 363609, 7257636, 11289600, 6234681600

Offset: 1

Giovanni Resta, Jul 19 2013

nonn

a(25), if it exists, is larger than 10^100.

			5832 = 18^3 = 4! + 4! + 4! + 6! + 7!.

Giovanni Resta, Decompositions of a(1)-a(24)

Cf. A114377, A227644-A227651.

A227647 Perfect powers equal to the sum of 6 factorial numbers.

Original entry on oeis.org

8, 9, 16, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 196, 256, 484, 729, 900, 1444, 1728, 2187, 2209, 2401, 5184, 5329, 5776, 5929, 7225, 7921, 11664, 15129, 20164, 41209, 45369, 46225, 80656, 121801, 126025, 201601, 363609, 443556, 776161, 806404

Offset: 1

Giovanni Resta, Jul 19 2013

nonn

a(51), if it exists, is larger than 10^100.

			443556 = 666^2 = 3! + 3! + 4! + 8! + 8! + 9!.

Giovanni Resta, Table of n, a(n) for n = 1..50 (terms < 10^100)
Giovanni Resta, Decompositions for a(1)-a(50)

Cf. A114377, A227644-A227651.

A227648 Perfect powers equal to the sum of 7 factorial numbers.

Original entry on oeis.org

8, 9, 16, 25, 27, 32, 36, 64, 81, 100, 128, 144, 169, 196, 256, 324, 484, 512, 529, 625, 729, 784, 841, 1089, 1225, 2187, 2197, 2916, 3025, 3721, 5184, 5776, 6241, 6724, 10816, 15129, 15876, 20164, 25921, 40401, 40804, 42025, 45369, 47524, 51984, 57600, 80656

Offset: 1

Giovanni Resta, Jul 19 2013

nonn

a(80), if it exists, is larger than 10^100.

			83521 = 17^4 = 1! + 6! + 6! + 6! + 6! + 8! + 8!.

Giovanni Resta, Table of n, a(n) for n = 1..79 (terms < 10^100)
Giovanni Resta, Decompositions for a(1)-a(79)

Cf. A114377, A227644-A227651.

A227649 Perfect powers equal to the sum of 8 factorial numbers.

Original entry on oeis.org

8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 343, 361, 400, 484, 512, 576, 729, 784, 841, 900, 961, 1000, 1089, 1521, 1600, 1849, 2304, 5041, 5184, 5776, 5832, 6724, 10201, 10816, 11881, 14400, 15129, 17424

Offset: 1

Giovanni Resta, Jul 19 2013

nonn

a(101), if it exists, is larger than 10^100.

			10890000 = 3300^2 = 6! + 6! + 6! + 6! + 6! + 10! + 10! + 10!.

Giovanni Resta, Table of n, a(n) for n = 1..100 (terms < 10^100)
Giovanni Resta, Decompositions for a(1)-a(100)

Cf. A114377, A227644-A227651.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A114377 Perfect powers equal to the sum of three factorial numbers.

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A227651 Perfect powers equal to the sum of 10 factorial numbers.

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A227650 Perfect powers equal to the sum of 9 factorial numbers.

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A227645 Perfect powers equal to the sum of 4 factorial numbers.

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A227646 Perfect powers equal to the sum of 5 factorial numbers.

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A227647 Perfect powers equal to the sum of 6 factorial numbers.

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A227648 Perfect powers equal to the sum of 7 factorial numbers.

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A227649 Perfect powers equal to the sum of 8 factorial numbers.

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

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