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

Author

Jonathan Vos Post, Sep 24 2006

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)

Formula

A000040 INTERSECTION A003353.

Extensions

More terms from Alois P. Heinz, Aug 12 2015