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A365008
Solutions k to the exponential Diophantine equation k^5 = Sum_{i=1..6} y_i^5 with positive y_i.
2
12, 24, 30, 32, 36, 48, 60, 64, 67, 72, 78, 84, 90, 96, 99, 106, 108, 112, 113, 119, 120, 128, 132, 134, 135, 139, 144, 145, 147, 150, 156, 160, 161, 168, 172, 178, 180, 189, 190, 192, 197, 198, 201, 202, 204, 205, 210, 212, 214, 216, 222, 223, 224, 225, 226, 227, 228, 234
OFFSET
1,1
COMMENTS
This includes nonprimitive solutions, where all {k, y_1, y_2, ..., y_6} have a common divisor > 1.
Primitive solutions are 12, 30, 32, 67, 78, 99, 106, 112, 113, 119, ... - Chai Wah Wu, Aug 18 2023
EXAMPLE
12, 24 and 60 are in the sequence as demonstrated by the following sets of solutions, [y_1,...,y_6] and k after a colon: [4, 5, 6, 7, 9, 11]: 12. [8, 10, 12, 14, 18, 22]: 24. [10, 20, 22, 32, 38, 58]: 60. [20, 25, 30, 35, 45, 55]: 60.
PROG
(Python)
from itertools import count, islice
from sympy import integer_nthroot
def A365008_gen(startvalue=1): # generator of terms >= startvalue
for n in count(max(startvalue, 1)):
n5, flag = n**5, False
for i1 in range(1, n):
i15=i1**5
for i2 in range(i1, n):
i25 = i15+i2**5
if i25>=n5: break
for i3 in range(i2, n):
i35 = i25+i3**5
if i35>=n5: break
for i4 in range(i3, n):
i45 = i35+i4**5
if i45>=n5: break
for i5 in range(i4, n):
i55 = i5**5
i65 = n5-i45-i55
if i65<i55: break
if integer_nthroot(i65, 5)[1]:
yield n
flag = True
break
if flag: break
if flag: break
if flag: break
if flag: break
A365008_list = list(islice(A365008_gen(), 4)) # Chai Wah Wu, Aug 17 2023
CROSSREFS
Sequence in context: A334665 A333945 A335146 * A182225 A074697 A333919
KEYWORD
nonn
AUTHOR
R. J. Mathar, Aug 16 2023
STATUS
approved