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A123038
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Prime sums of 9 positive 5th powers.
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1
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71, 251, 257, 313, 499, 617, 797, 859, 977, 1039, 1063, 1187, 1249, 1367, 1429, 1523, 1609, 1789, 1913, 2179, 2273, 2297, 2539, 2663, 2843, 3023, 3109, 3257, 3319, 3413, 3499, 3593, 3617, 3803, 4373, 4733, 4889, 5179, 5303, 5483, 5639, 5881, 6257, 6389, 6451
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 71 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5.
a(2) = 251 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(3) = 257 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 313 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5.
a(5) = 499 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5
a(9) = 977 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5 + 3^5 + 3^5.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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