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A123037
Prime sums of 8 positive 5th powers.
1
101, 163, 281, 467, 523, 647, 827, 1031, 1069, 1093, 1217, 1249, 1459, 1733, 1999, 2389, 3163, 3319, 3467, 3529, 3623, 3709, 3803, 3889, 4217, 4373, 4397, 4639, 4943, 5209, 5333, 5693, 5849, 6263, 6287, 6529, 6653, 6833, 7013, 7411, 7583, 7907, 8087, 8329
OFFSET
1,1
COMMENTS
Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.
There must be an odd number of odd terms in the sum, either one even and seven odd (as with 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5 and 523 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5), three even and 5 odd terms (as with 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5), five even and 3 odd terms (as with 647 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5) or seven even terms and one odd term (as with 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.
LINKS
FORMULA
A000040 INTERSECTION A003353.
EXAMPLE
a(1) = 101 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5.
a(2) = 163 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(3) = 281 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5.
a(4) = 467 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5.
a(5) = 523 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5.
a(6) = 647 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.
MATHEMATICA
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, 8}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Sep 24 2006
EXTENSIONS
More terms from Alois P. Heinz, Aug 12 2015
STATUS
approved