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%I #4 Jun 02 2017 00:30:01
%S 1,2,4,6,3,7,8,10,12,11,9,16,14,18,28,20,22,32,33,13,24,38,30,36,34,
%T 26,42,48,40,44,46,50,60,52,68,54,58,5,15,64,78,56,66,70,74,76,84,62,
%U 72,82,90,80,55,21,92,106,104,88,98,100,96,108,102,86,114,94,116,118,122,120,130,126,107,31,132,138
%N a(n) is the smallest positive integer not already in sequence such that a(1) + ... + a(n) is a prime power, with a(1) = 1.
%C It appears that the sequence contains all even numbers.
%e a(8) = 10 because 1, 2, 3, 4, 6, 7 and 8 have already been used in the sequence, 1 + 2 + 4 + 6 + 3 + 7 + 8 + 5 = 36 is not prime power, 1 + 2 + 4 + 6 + 3 + 7 + 8 + 9 = 40 is not prime power while 1 + 2 + 4 + 6 + 3 + 7 + 8 + 10 = 41 is a prime power.
%t t = {1}; Do[i = 1; While[! PrimePowerQ[Total[t] + i] || MemberQ[t, i], i++]; AppendTo[t, i], {75}]; t
%Y Cf. A000961, A054408, A073659, A073879, A075562, A084693, A284049.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, May 29 2017