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%I #11 Jun 13 2016 04:18:24
%S 71,251,257,313,499,617,797,859,977,1039,1063,1187,1249,1367,1429,
%T 1523,1609,1789,1913,2179,2273,2297,2539,2663,2843,3023,3109,3257,
%U 3319,3413,3499,3593,3617,3803,4373,4733,4889,5179,5303,5483,5639,5881,6257,6389,6451
%N Prime sums of 9 positive 5th powers.
%C Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.
%C There must be an odd number of odd terms in the sum, either nine odd (as with 251 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 and 977 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5 + 3^5 + 3^5), two even and seven odd (as with 71 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 and 313 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5), four even and 5 odd terms (as with xxxx), six even and 3 odd terms (as with 3803 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 5^5) or eight even terms and one odd term (as with 257 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 and 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.
%H Giovanni Resta, <a href="/A123038/b123038.txt">Table of n, a(n) for n = 1..10000</a>
%F A000040 INTERSECTION A003354.
%e a(1) = 71 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5.
%e a(2) = 251 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
%e a(3) = 257 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
%e a(4) = 313 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5.
%e a(5) = 499 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5
%e a(9) = 977 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5 + 3^5 + 3^5.
%t up = 10^4; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 9}]; Select[a, PrimeQ] (* _Giovanni Resta_, Jun 13 2016 *)
%Y Cf. A000040, A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Sep 24 2006
%E More terms from _Alois P. Heinz_, Aug 12 2015
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