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%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