login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Prime sums of 12 positive 5th powers.
1

%I #12 Aug 11 2015 10:36:29

%S 43,167,229,347,353,409,769,1097,1277,1283,1439,1619,1823,1861,1979,

%T 2003,2089,2213,2221,2393,2549,2579,2729,2791,2939,2971,3001,3119,

%U 3167,3181,3229,3299,3323,3329,3361,3533,3541,3571,3697,3931,4049,4079,4111,4159,4259

%N Prime sums of 12 positive 5th powers.

%C Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}. There must be an odd number of odd terms in the sum, either one even and 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 + 2^5 and 769 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5 + 3^5), three even and nine odd (as with 347 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5), five even and seven odd (as with 167 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 and 409 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5), seven even and 5 odd terms (as with 229 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5), nine even and 3 odd terms (as with 161341 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 11^5) or eleven even terms and one odd term (as with 353 = 1^ 5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.

%H Matthew House, <a href="/A123040/b123040.txt">Table of n, a(n) for n = 1..10000</a>

%F A000040 INTERSECTION A003357.

%e a(1) = 43 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.

%e a(2) = 167 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.

%e a(3) = 229 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.

%e a(4) = 347 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5.

%e a(5) = 353 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.

%e a(6) = 409 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5.

%e a(7) = 769 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5 + 3^5.

%p N:= 10000: # to get all terms <= N

%p B:= {seq(i^5,i=1..floor(N^(1/5)))}:

%p B2:= select(`<=`,{seq(seq(b+c,b=B),c=B)},N):

%p B4:= select(`<=`,{seq(seq(b+c,b=B2),c=B2)},N):

%p B8:= select(`<=`,{seq(seq(b+c,b=B4),c=B4)},N):

%p B12:= select(`<=`,{seq(seq(b+c,b=B4),c=B8)},N):

%p sort(select(isprime,convert(B12,list))); # _Robert Israel_, Aug 10 2015

%Y Cf. A000040, A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Sep 24 2006

%E More terms from _Matthew House_, Aug 10 2015