A331674 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has at least two primitive solutions in nonnegative integers.

744, 1686, 1921, 2087, 3447, 4097, 6065, 7157, 7864, 8570

Offset

1,1

Comments

Primitive solutions means gcd(a,b,c,d,e) = 1.

These are all terms from James Waldby link, which gives all solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 where k < 10000, gcd(a,b,c,d,e) = 1 and at least two of a,b,c,d,e are nonzero.

Note that if nonprimitive solutions were allowed (where at least two of a,b,c,d,e are nonzero), then 144 would be a term because 144^5 = 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 38^5 + 86^5 + 92^5 + 94^5 + 134^5.

Links

Table of n, a(n) for n=1..10.

James Waldby, A Table of Fifth Powers equal to a Fifth Power

Example

Solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 = a'^5 + b'^5 + c'^5 + d'^5 + e'^5:
744: (100, 210, 414, 629, 651), (14, 95, 545, 586, 644);
1686: (265, 486, 784, 791, 1670), (46, 591, 675, 999, 1655);
1921: (275, 351, 872, 1298, 1855), (95, 771, 1020, 1519, 1756);
2087: (145, 565, 1105, 1462, 1990), (519, 642, 1026, 1480, 1990);
3447: (1212, 1300, 1345, 1699, 3411), (289, 317, 1033, 1682, 3426);
4097: (1281, 2154, 2396, 3462, 3504), (954, 1989, 2127, 2396, 3981);
6065: (3629, 3811, 4070, 4272, 5313), (854, 3160, 3752, 5073, 5196);
7157: (1827, 2186, 4789, 5629, 6376), (930, 2746, 3570, 5109, 6802);
7864: (1093, 2309, 3629, 6137, 7296), (312, 1631, 3418, 3544, 7809);
8570: (1766, 2529, 4086, 5520, 8319), (2101, 2315, 2710, 3960, 8524).

Crossrefs

Subsequence of A063923 (and thus of A063922).

Other similar sequences:

A023041 (k^3=a^3+b^3+c^3, gcd(a,b,c)=1);

A003828 (k^4=a^4+b^4+c^4, gcd(a,b,c)=1);

A175610 (k^4=a^4+b^4+c^4);

A039664 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1);

A003294 (k^4=a^4+b^4+c^4+d^4);

A331675 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1, at least two solutions).

A134341 (k^5=a^5+b^5+c^5+d^5).

Sequence in context: A345800 A331092 A235294 * A235252 A268891 A306281

Adjacent sequences: A331671 A331672 A331673 * A331675 A331676 A331677

Keyword

nonn,hard,more

Author

Jianing Song, Jan 24 2020

Status

approved