A123040 Prime sums of 12 positive 5th powers.

43, 167, 229, 347, 353, 409, 769, 1097, 1277, 1283, 1439, 1619, 1823, 1861, 1979, 2003, 2089, 2213, 2221, 2393, 2549, 2579, 2729, 2791, 2939, 2971, 3001, 3119, 3167, 3181, 3229, 3299, 3323, 3329, 3361, 3533, 3541, 3571, 3697, 3931, 4049, 4079, 4111, 4159, 4259

Offset

1,1

Comments

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.

Links

Matthew House, Table of n, a(n) for n = 1..10000

Formula

A000040 INTERSECTION A003357.

Example

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.
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.
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.
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.
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.
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.
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.

Maple

N:= 10000: # to get all terms <= N
B:= {seq(i^5, i=1..floor(N^(1/5)))}:
B2:= select(`<=`, {seq(seq(b+c, b=B), c=B)}, N):
B4:= select(`<=`, {seq(seq(b+c, b=B2), c=B2)}, N):
B8:= select(`<=`, {seq(seq(b+c, b=B4), c=B4)}, N):
B12:= select(`<=`, {seq(seq(b+c, b=B4), c=B8)}, N):
sort(select(isprime, convert(B12, list))); # Robert Israel, Aug 10 2015

Crossrefs

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

Sequence in context: A078852 A078956 A206399 * A142016 A140640 A083357

Adjacent sequences: A123037 A123038 A123039 * A123041 A123042 A123043

Keyword

easy,nonn

Author

Jonathan Vos Post, Sep 24 2006

Extensions

More terms from Matthew House, Aug 10 2015

Status

approved