login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Solutions k to the exponential Diophantine equation k^5 = Sum_{i=1..6} y_i^5 with positive y_i.
2

%I #42 Aug 24 2023 02:42:25

%S 12,24,30,32,36,48,60,64,67,72,78,84,90,96,99,106,108,112,113,119,120,

%T 128,132,134,135,139,144,145,147,150,156,160,161,168,172,178,180,189,

%U 190,192,197,198,201,202,204,205,210,212,214,216,222,223,224,225,226,227,228,234

%N Solutions k to the exponential Diophantine equation k^5 = Sum_{i=1..6} y_i^5 with positive y_i.

%C This includes nonprimitive solutions, where all {k, y_1, y_2, ..., y_6} have a common divisor > 1.

%C Primitive solutions are 12, 30, 32, 67, 78, 99, 106, 112, 113, 119, ... - _Chai Wah Wu_, Aug 18 2023

%H Chai Wah Wu, <a href="/A365008/b365008.txt">Table of n, a(n) for n = 1..118</a>

%H <a href="/index/Di#Diophantine">Index to sequences related to Diophantine equations</a> (5,1,6).

%e 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.

%o (Python)

%o from itertools import count, islice

%o from sympy import integer_nthroot

%o def A365008_gen(startvalue=1): # generator of terms >= startvalue

%o for n in count(max(startvalue,1)):

%o n5, flag = n**5, False

%o for i1 in range(1,n):

%o i15=i1**5

%o for i2 in range(i1,n):

%o i25 = i15+i2**5

%o if i25>=n5: break

%o for i3 in range(i2,n):

%o i35 = i25+i3**5

%o if i35>=n5: break

%o for i4 in range(i3,n):

%o i45 = i35+i4**5

%o if i45>=n5: break

%o for i5 in range(i4,n):

%o i55 = i5**5

%o i65 = n5-i45-i55

%o if i65<i55: break

%o if integer_nthroot(i65,5)[1]:

%o yield n

%o flag = True

%o break

%o if flag: break

%o if flag: break

%o if flag: break

%o if flag: break

%o A365008_list = list(islice(A365008_gen(),4)) # _Chai Wah Wu_, Aug 17 2023

%K nonn

%O 1,1

%A _R. J. Mathar_, Aug 16 2023