A114377 -id:A114377 - cp's OEIS frontend

A227644 Perfect powers equal to the sum of 2 factorial numbers.

4, 8, 25, 121, 144, 5041

Offset: 1

Giovanni Resta, Jul 19 2013

a(7), if it exists, is greater than 10^100.
a(7), if it exists, is greater than 10000!. - Filip Zaludek, Jul 18 2017
a(7), if it exists, is greater than 11750!. - Filip Zaludek, Sep 07 2018
a(7), if it exists, is greater than 20000!. - Filip Zaludek, Nov 04 2020

			5041 = 71^2 = 1! + 7!.

Giovanni Resta, Decompositions of a(1)-a(6)

Cf. A114377, A227645, A227646, A227647, A227648, A227649, A227650, A227651, A065433, A085692.

/* To compile: gcc -Wall -O2 A227644.c -o A227644 -lgmp */
#include 
#include 
#include 
int main()
{
   int bsz=256, a=0;
   mpz_t *f, t;
   f = malloc(sizeof(mpz_t) * bsz);
   mpz_init(t); mpz_init(f[0]); mpz_set_ui(f[0], 1);
   while (1)
   {
       a += 1;
       if (a == bsz)
       {
           bsz *= 2;
           f = (mpz_t *) realloc(f, sizeof(mpz_t) * bsz);
       }
       mpz_init(f[a]);
       mpz_mul_ui(f[a], f[a-1], a);
       for (int i=1; i<=a; i++)
       {
           mpz_add(t, f[a], f[i]);
           if (mpz_perfect_power_p(t))
           {
               gmp_printf("%Zd, ", t);
               fflush(stdout);
           }
       }
   }
   return 0;
}

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.

A082875 Squares that are the sum of three factorials.

Original entry on oeis.org

4, 9, 36, 49, 841, 5184

Offset: 1

Cino Hilliard, May 25 2003

			These appear to be the only solutions. 8 and 27 appear to be the only cubes that are the sum of 3 factorials. Again, it appears that 2 and 3 are the only powers of n satisfying a1!+a2!+a3! = z^n.
The complete list of solutions is
a1 a2 a3 z^2
0 0 2 4
0 1 2 4
0 2 3 9
0 4 4 49
0 5 6 841
1 1 2 4
1 2 3 9
1 4 4 49
1 5 6 841
3 3 4 36
4 5 7 5184

Cf A114377, A162681.

d = 50; a = Union[ Flatten[ Table[a! + b! + c!, {a, 1, d}, {b, a, d}, {c, b, d}]]]; l = Length[a]; Do[ If[ IntegerQ[ Sqrt[ a[[i]]]], Print[ a[[i]]]], {i, 1, l}]

sum3factsq(n) = { for(a1=1,n, for(a2=a1,n, for(a3=a2,n, z = a1!+a2!+a3!; if(issquare(z),print1(z" ")) ) ) ) }

a1! + a2! + a3! = z^2.

Sequence data ordered by Michel Marcus, Jun 03 2013

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.

cp's OEIS Frontend

A227644 Perfect powers equal to the sum of 2 factorial numbers.

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

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A082875 Squares that are the sum of three factorials.

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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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