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A123039
Prime sums of 11 positive 5th powers.
1
11, 73, 197, 439, 557, 563, 619, 743, 1103, 1283, 1307, 1493, 1549, 2243, 2251, 2399, 2423, 2579, 2969, 3001, 3259, 3329, 3391, 3539, 3571, 3719, 3923, 4079, 4289, 4493, 4649, 4673, 5039, 5281, 5399, 5641, 5851, 6211, 6359, 6367, 6421, 6563, 6719, 6781, 6961
OFFSET
1,1
COMMENTS
Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.
There must be an odd number of odd terms in the sum, either eleven odd (as with 11 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5), two even and nine odd (as with 73 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 and 557 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5 + 3^5), four even and seven odd (as with 619 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5), six even and 5 odd terms (as with 197 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 and 439 = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5), eight even and 3 odd terms (as with 743 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5) or ten even terms and one odd term (as with 2^5 + 2^5 + 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.
LINKS
FORMULA
A000040 INTERSECTION A003356.
EXAMPLE
a(1) = 11 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 73 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5.
a(3) = 197 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 439 = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5.
a(5) = 557 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5 + 3^5.
a(6) = 563 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5.
a(7) = 619 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.
a(8) = 743 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.
MATHEMATICA
up = 6961; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 11}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 12 2016 *)
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Sep 24 2006
EXTENSIONS
More terms from Alois P. Heinz, Aug 12 2015
STATUS
approved