A003347 -id:A003347 - cp's OEIS frontend

A085319 Primes which are the sum of three 5th powers.

3, 307, 487, 9043, 16871, 17293, 17863, 23057, 32359, 32801, 33857, 36739, 40787, 43669, 50599, 59051, 59113, 62417, 65537, 76099, 101267, 104149, 107777, 135893, 160073, 161053, 164419, 249107, 249857, 256609, 259733, 266663, 338909, 340649

Offset: 1

Labos Elemer, Jul 01 2003

nonn

Primes in the sumset {A000584 + A000584 + A000584}. There must be an odd number of odd terms in the sum, either 3 odd terms (as with 3 = 1^5 + 1^5 + 1^5 and 487 = 1^5 + 3^5 + 3^5 and 59051 = 1^5 + 1^5 + 9^5) or two even terms and one odd term (as with 307 = 2^5 + 2^5 + 3^5 and 9043 = 3^5 + 4^5 + 6^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime. - Jonathan Vos Post, Sep 24 2006

			a(1) = 3 = 1^5 + 1^5 + 1^5.
a(2) = 307 = 2^5 + 2^5 + 3^5.
a(3) = 487 = 1^5 + 3^5 + 3^5.
a(4) = 9043 = 3^5 + 4^5 + 6^5.
a(5) = 16871 = 2^5 + 2^5 + 7^5.
a(6) = 17293 = 3^5 + 3^5 + 7^5.

Vladimir Joseph Stephan Orlovsky, Table of n, a(n) for n = 1..1000

Cf. A003348, A007490, A085318.

Cf. A000040, A000584, A003336, A003347.

lim = 10^6; nn = Floor[(lim - 2)^(1/5)]; t = {}; Do[p = i^5 + j^5 + k^5; If[p <= lim && PrimeQ[p], AppendTo[t, p]], {i, nn}, {j, i}, {k, j}]; t = Union[t] (* Vladimir Joseph Stephan Orlovsky and T. D. Noe, Jul 15 2011 *)
Select[Prime[Range[2,30000]],Length[PowersRepresentations[#,3,5]]>0&] (* Harvey P. Dale, Nov 26 2014 *)

A123032 was identical. - T. D. Noe, Jul 15 2011

A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573

Offset: 1

N. J. A. Sloane

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

A###### (x, y): Numbers that are the form of x nonzero y-th powers.

Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

b-file checked by R. J. Mathar, Aug 01 2020

Incorrect program removed by David A. Corneth, Aug 01 2020

A123294 Sum of 13 positive 5th powers.

Original entry on oeis.org

13, 44, 75, 106, 137, 168, 199, 230, 255, 261, 286, 292, 317, 323, 348, 354, 379, 385, 410, 416, 441, 472, 497, 503, 528, 534, 559, 565, 590, 596, 621, 627, 652, 683, 714, 739, 745, 770, 776, 801, 807, 832, 838, 863, 894, 925, 956, 981, 987, 1012, 1018, 1036

Offset: 1

Jonathan Vos Post, Sep 24 2006

Up to 416 = 13*(2^5) this sequence is identical to x+1 for x in A003357 Sum of 12 positive 5th powers. Primes in this sequence (13, 137, 199, 317, ...) are A123299. As proved by J.-R. Chen in 1964, g(5) = 37, so every positive integer can be written as the sum of no more than 37 positive 5th powers. G(5) <= 17, bounding the least integer G(5) such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(5) 5th powers.

			a(1) = 13 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 44 = 1^5 + 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(9) = 255 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(11) = 286 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123295-A123297, A008480, A123299.

up = 1500; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 13}]; a (* Giovanni Resta, Jun 12 2016 *)

Sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

Two missing terms and more terms from Giovanni Resta, Jun 12 2016

A123295 Sum of 14 positive 5th powers.

Original entry on oeis.org

14, 45, 76, 107, 138, 169, 200, 231, 256, 262, 287, 293, 318, 324, 349, 355, 380, 386, 411, 417, 442, 448, 473, 498, 504, 529, 535, 560, 566, 591, 597, 622, 628, 653, 659, 684, 715, 740, 746, 771, 777, 802, 808, 833, 839, 864, 870, 895, 926, 957, 982, 988

Offset: 1

Jonathan Vos Post, Sep 24 2006

Up to 417 = 13*(2^5) + 1 this sequence is identical to x+2 for x in A003357 Sum of 12 positive 5th powers. Primes in this sequence (107, 293, 349, 653, ...) are A123300. As proved by J.-R. Chen in 1964, g(5) = 37, so every positive integer can be written as the sum of no more than 37 positive 5th powers. G(5) <= 17, bounding the least integer G(5) such that every positive integer beyond a certain point (i.e., all but a finite number) is the sum of G(5) 5th powers.

			a(1) = 14 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 45 = 1^5 + 1^5 + 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(9) = 256 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(11) = 287 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123294-A123297, A008480, A123299, A123300.

up = 1000; q = Range[up^(1/5)]^5; a ={0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a=b, {k, 14}]; a (* Giovanni Resta, Jun 12 2016 *)

Sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

5 missing terms and more terms from Giovanni Resta, Jun 12 2016

A123299 Prime sums of 13 positive 5th powers.

Original entry on oeis.org

13, 137, 199, 317, 379, 503, 683, 739, 863, 1049, 1129, 1223, 1229, 1409, 1433, 1471, 1613, 1619, 1831, 1949, 1979, 2011, 2221, 2339, 2543, 2549, 2729, 2791, 2909, 2917, 2971, 3089, 3137, 3299, 3307, 3323, 3331, 3361, 3511, 3541, 3659, 3863, 3877, 3931, 4049

Offset: 1

Jonathan Vos Post, Sep 24 2006

			a(1) = 13 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 137 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(3) = 199 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 317 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5.

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

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123294, A123295, A008480.

up = 4100; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 13}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 12 2016 *)

A000040 INTERSECTION A123299.

a(10)-a(45) from Giovanni Resta, Jun 12 2016

A123300 Prime sums of 14 positive 5th powers.

Original entry on oeis.org

107, 293, 349, 653, 659, 839, 1013, 1019, 1223, 1279, 1409, 1559, 1583, 1621, 1801, 1831, 1949, 2011, 2129, 2153, 2309, 2333, 2339, 2347, 2371, 2551, 2699, 2707, 2731, 2879, 2917, 3083, 3121, 3169, 3191, 3301, 3331, 3449, 3457, 3511, 3541, 3659, 3691, 3761, 3847, 4019, 4027, 4051

Offset: 1

Jonathan Vos Post, Sep 25 2006

			a(1) = 107 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5.
a(2) = 293 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(3) = 349 = 1^5 + 1^5 + 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(4) = 653 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.

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

Cf. A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356, A003357, A123294-A123299.

up = 5000; q = Range[up^(1/5)]^5; a={0}; Do[b = Select[Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 14}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 12 2016 *)

A000040 INTERSECTION A123295.

More terms from Harvey P. Dale, Jan 01 2015

4 missing terms from Giovanni Resta, Jun 12 2016

A123033 Prime sums of 4 positive 5th powers.

Original entry on oeis.org

97, 277, 761, 1511, 1753, 2081, 3221, 3643, 6197, 7517, 7841, 8263, 10067, 10399, 10903, 16903, 25639, 32771, 32833, 33013, 33647, 33889, 35059, 36137, 39019, 40577, 40819, 48563, 49639, 57383, 59083, 59567, 60317, 61129, 62207, 63199, 66383, 66889, 100003

Offset: 1

Jonathan Vos Post, Sep 24 2006

Primes in the sumset {A000584 + A000584 + A000584 + A000584}.

There must be an odd number of odd terms in the sum, either one even and 3 odd terms (as with 1^5 + 1^5 + 2^5 + 3^5 and 761 = 2^5 + 3^5 + 3^5 + 3^5) or three even terms and one odd term (as with 97 = 1^5 + 2^5 + 2^5 + 2^5 and 3221 = 2^5 + 2^5 + 2^5 + 5^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.

			a(1) = 97 = 1^5 + 2^5 + 2^5 + 2^5.
a(2) = 277 = 1^5 + 1^5 + 2^5 + 3^5.
a(3) = 761 = 2^5 + 3^5 + 3^5 + 3^5.
a(7) = 3221 = 2^5 + 2^5 + 2^5 + 5^5.

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

Cf. A000040, A000584, A003336, A003347, A003349.

up = 10^6; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@Table[e + a, {e, q}], # <= up &]; a = b, {k, 4}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)

A000040 INTERSECTION A003349.

More terms from Alois P. Heinz, Aug 12 2015

A123034 Prime sums of 5 positive 5th powers.

Original entry on oeis.org

5, 67, 1301, 1543, 2113, 2293, 2777, 3191, 3253, 3347, 3371, 3433, 3613, 4339, 5237, 5417, 5659, 6229, 6737, 7307, 7549, 7873, 8053, 8537, 8803, 9377, 9439, 9619, 9857, 10099, 11177, 11423, 11927, 12743, 15797, 15859, 16811, 17053, 17183, 18679, 18919, 19163

Offset: 1

Jonathan Vos Post, Sep 24 2006

Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584}.

There must be an odd number of odd terms in the sum, either 5 odd terms (as with 5 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 and 16811 = 1^5 + 1^5 + 1^5 + 1^5 + 7^5), two even and 3 odd terms (as with 67 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 and 1301 = 1^5 + 1^5 + 2^5 + 3^5 + 4^5) or four even terms and one odd term (as with 3253 = 2^5 + 2^5 + 2^5 + 2^5 + 5^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.

			a(1) = 5 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 67 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5.
a(3) = 1301 = 1^5 + 1^5 + 2^5 + 3^5 + 4^5.
a(4) = 1543 = 1^5 + 2^5 + 3^5 + 3^5 + 4^5.
a(5) = 2113 = 1^5 + 2^5 + 2^5 + 4^5 + 4^5.
a(6) = 3191 = 1^5 + 1^5 + 2^5 + 2^5 + 5^5.
a(7) = 4339 = 3^5 + 4^5 + 4^5 + 4^5 + 4^5.

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

Cf. A000040, A000584, A003336, A003347, A003349, A003350.

Take[Select[Union[Total/@Tuples[Range[10]^5,5]],PrimeQ],60] (* Harvey P. Dale, Jul 21 2014 *)

A000040 INTERSECTION A003350.

Corrected and extended by Harvey P. Dale, Jul 21 2014

A123035 Prime sums of 6 positive 5th powers.

Original entry on oeis.org

37, 521, 1091, 1153, 1997, 2083, 2239, 3137, 3559, 4129, 4153, 4457, 4637, 5449, 6199, 7253, 8147, 8573, 9319, 9323, 10069, 10463, 11959, 14029, 15083, 15649, 16649, 16843, 16883, 17327, 17389, 17569, 17959, 18077, 18773, 18803, 19373, 20029

Offset: 1

Jonathan Vos Post, Sep 24 2006

nonn

Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

There must be an odd number of odd terms in the sum, either one even and 5 odd terms (as with 37 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 and 521 = 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5), three even and 3 odd terms (as with 1091 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 4^5) or five even terms and one odd term (as with 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 4^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.

			a(1) = 37 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
a(2) = 521 = 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5.
a(3) = 1091 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 4^5.
a(4) = 1153 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 4^5.

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

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

up = 10^6; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@Table[e + a, {e, q}], # <= up &]; a = b, {k, 6}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)

A000040 INTERSECTION A003351.

More terms from Max Alekseyev, Sep 24 2011

A123036 Prime sums of 7 positive 5th powers.

Original entry on oeis.org

7, 131, 193, 311, 373, 491, 733, 857, 1061, 1123, 1217, 1279, 1303, 1427, 1459, 1607, 1787, 2029, 2053, 2357, 3169, 3373, 3677, 3739, 3833, 3919, 4099, 4583, 5153, 5419, 5903, 6317, 6379, 6473, 7043, 7309, 7793, 7937, 8117, 8179, 8297, 8363, 8539, 8543, 8867

Offset: 1

Jonathan Vos Post, Sep 24 2006

Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

There must be an odd number of odd terms in the sum, either seven odd (as with 7 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5), two even and 5 odd terms (as with 311 = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5), four even and 3 odd terms (as with 131 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 and 373 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5) or six even terms and one odd term (as with 193 = 1^5 + 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.

			a(1) = 7 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 131 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(3) = 193 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 311 = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5.
a(5) = 373 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5.

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

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

Take[Union[Select[Total/@Tuples[Range[8]^5,7],PrimeQ]],50] (* Harvey P. Dale, May 08 2012 *)

A000040 INTERSECTION A003352.

More terms from Harvey P. Dale, May 08 2012

cp's OEIS Frontend

A085319 Primes which are the sum of three 5th powers.

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A003386 Numbers that are the sum of 8 nonzero 8th powers.

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A123294 Sum of 13 positive 5th powers.

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A123295 Sum of 14 positive 5th powers.

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A123299 Prime sums of 13 positive 5th powers.

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A123300 Prime sums of 14 positive 5th powers.

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A123033 Prime sums of 4 positive 5th powers.

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