A123037 Prime sums of 8 positive 5th powers.

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

Giovanni Resta, Table of n, a(n) for n = 1..10000

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 *)

Crossrefs

Cf. A000040, A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353.

Sequence in context: A243365 A107186 A142660 * A142012 A067860 A140037

Adjacent sequences: A123034 A123035 A123036 * A123038 A123039 A123040

Keyword

easy,nonn

Author

Jonathan Vos Post, Sep 24 2006

Extensions

More terms from Alois P. Heinz, Aug 12 2015

Status

approved