A320249 Numbers not of the form [p^c] + [q^c] + [r^c] + [s^c] + [t^c] where p, q, r, s, and t are prime, c = 41/20 = 2.05, and [...] is the floor function.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 44, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 72, 73, 74, 77, 78, 79, 82, 83, 84, 86, 87, 88, 91, 92, 95, 96

Offset

1,2

Comments

Zhang & Li prove that this sequence is finite. More generally, for any 1 < c < 11216182/5471123 = 2.0500694... except c = 2, there are only finitely many numbers not of the form [p^c] + [q^c] + [r^c] + [s^c] + [t^c] where p, q, r, s, and t are prime.

It seems that a(1058) = 15980 is the last term. If there are any further terms they are larger than 7 * 10^12. - Charles R Greathouse IV, Oct 08 2018

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..1058

Min Zhang and Jinjiang Li, On a Diophantine equation with five prime variables, arXiv:1809.04591 [math.NT], 2018.

Example

Missing:
20 = [2^2.05] + [2^2.05] + [2^2.05] + [2^2.05] + [2^2.05]
25 = [2^2.05] + [2^2.05] + [2^2.05] + [2^2.05] + [3^2.05]
30 = [2^2.05] + [2^2.05] + [2^2.05] + [3^2.05] + [3^2.05]
35 = [2^2.05] + [2^2.05] + [3^2.05] + [3^2.05] + [3^2.05]
40 = [2^2.05] + [3^2.05] + [3^2.05] + [3^2.05] + [3^2.05]
43 = [2^2.05] + [2^2.05] + [2^2.05] + [2^2.05] + [5^2.05]
45 = [3^2.05] + [3^2.05] + [3^2.05] + [3^2.05] + [3^2.05]

Crossrefs

Sequence in context: A347520 A180493 A346132 * A309066 A084981 A326880

Adjacent sequences: A320246 A320247 A320248 * A320250 A320251 A320252

Keyword

nonn,fini

Author

Charles R Greathouse IV, Oct 08 2018

Status

approved