A331675 Numbers k such that k^4 = a^4 + b^4 + c^4 + d^4 has at least two positive primitive solutions.

31127, 41963, 72899, 154789, 195479, 208471

Offset

1,1

Comments

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

These are all terms from Jaroslaw Wroblewski link, which gives all positive solutions to k^4 = a^4 + b^4 + c^4 + d^4 where k < 222000, gcd(a,b,c,d) = 1.

Links

Table of n, a(n) for n=1..6.

Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000 (Note: (t,m,n) denotes the equation Sum_{i=1..m} (a_i)^t = Sum_{j=1..n} (b_j)^t, where a_i, b_j are positive integers, gcd(a_1, a_2, ..., a_m, b_1, b_2, ..., b_n) = 1.)

Example

Solutions to k^4 = a^4 + b^4 + c^4 + d^4 = a'^4 + b'^4 + c'^4 + d'^4:
31127: (2260, 4870, 17386, 30335), (2495, 11998, 16430, 30320);
41963: (1100, 17260, 25015, 40234), (8750, 12109, 14470, 41720);
72899: (4555, 44270, 58868, 59330), (9700, 16480, 47618, 69265);
154789: (49586, 55450, 102170, 145615), (66405, 106740, 119760, 121664);
195479: (12970, 43340, 140947, 180520), (25570, 41080, 112822, 189695);
208471: (3903, 46560, 61290, 207950), (91045, 149222, 150550, 168730).

Crossrefs

Subsequence of A039664 (and thus of A003294).

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);

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

A063923 (k^5=a^5+b^5+c^5+d^5+e^5, gcd(a,b,c,d,e)=1);

A063922 (k^5=a^5+b^5+c^5+d^5+e^5);

A331674 (k^5=a^5+b^5+c^5+d^5+e^5, gcd(a,b,c,d,e)=1, at least two solutions).

Sequence in context: A090059 A234888 A151815 * A285695 A186800 A230753

Adjacent sequences: A331672 A331673 A331674 * A331676 A331677 A331678

Keyword

nonn,hard,more

Author

Jianing Song, Jan 24 2020

Status

approved