A003347 -id:A003347 - cp's OEIS frontend

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

Jonathan Vos Post, Sep 24 2006

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.

			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.

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

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

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 *)

A000040 INTERSECTION A003353.

More terms from Alois P. Heinz, Aug 12 2015

A123038 Prime sums of 9 positive 5th powers.

Original entry on oeis.org

71, 251, 257, 313, 499, 617, 797, 859, 977, 1039, 1063, 1187, 1249, 1367, 1429, 1523, 1609, 1789, 1913, 2179, 2273, 2297, 2539, 2663, 2843, 3023, 3109, 3257, 3319, 3413, 3499, 3593, 3617, 3803, 4373, 4733, 4889, 5179, 5303, 5483, 5639, 5881, 6257, 6389, 6451

Offset: 1

Jonathan Vos Post, Sep 24 2006

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 nine odd (as with 251 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 and 977 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5 + 3^5 + 3^5), two even and seven odd (as with 71 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 and 313 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5), four even and 5 odd terms (as with xxxx), six even and 3 odd terms (as with 3803 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 5^5) or eight even terms and one odd term (as with 257 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 and 2^5 + 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.

			a(1) = 71 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5.
a(2) = 251 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5.
a(3) = 257 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 313 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5.
a(5) = 499 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5
a(9) = 977 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5 + 3^5 + 3^5.

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

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

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, 9}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)

A000040 INTERSECTION A003354.

More terms from Alois P. Heinz, Aug 12 2015

A123039 Prime sums of 11 positive 5th powers.

Original entry on oeis.org

11, 73, 197, 439, 557, 563, 619, 743, 1103, 1283, 1307, 1493, 1549, 2243, 2251, 2399, 2423, 2579, 2969, 3001, 3259, 3329, 3391, 3539, 3571, 3719, 3923, 4079, 4289, 4493, 4649, 4673, 5039, 5281, 5399, 5641, 5851, 6211, 6359, 6367, 6421, 6563, 6719, 6781, 6961

Offset: 1

Jonathan Vos Post, Sep 24 2006

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

There must be an odd number of odd terms in the sum, either 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), two even and nine odd (as with 73 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 and 557 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5 + 3^5), four even and seven odd (as with 619 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5), six even and 5 odd terms (as with 197 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 and 439 = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5), eight even and 3 odd terms (as with 743 = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5) or ten even terms and one odd term (as with 2^5 + 2^5 + 2^5 + 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.

			a(1) = 11 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5.
a(2) = 73 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5.
a(3) = 197 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 439 = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5.
a(5) = 557 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 3^5 + 3^5.
a(6) = 563 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5.
a(7) = 619 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.
a(8) = 743 = 1^5 + 2^5 + 2^5 + 2^5 + 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. A000040, A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355, A003356.

up = 6961; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 11}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 12 2016 *)

A000040 INTERSECTION A003356.

More terms from Alois P. Heinz, Aug 12 2015

A123040 Prime sums of 12 positive 5th powers.

Original entry on oeis.org

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

Jonathan Vos Post, Sep 24 2006

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.

			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.

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

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

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

A000040 INTERSECTION A003357.

More terms from Matthew House, Aug 10 2015

A123043 Prime sums of 10 positive 5th powers.

Original entry on oeis.org

41, 103, 227, 283, 587, 829, 953, 1009, 1033, 1399, 1493, 1523, 1579, 1759, 2063, 2087, 2243, 2273, 2633, 2789, 2969, 3079, 3203, 3359, 3407, 3413, 3469, 3539, 3593, 3929, 4133, 4157, 4219, 4289, 4523, 4679, 4703, 5101, 5273, 5851, 6203, 6389, 6421, 6569, 6991

Offset: 1

Jonathan Vos Post, Sep 24 2006

Primes in the sumset {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 nine odd (as with 41 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 and 283 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5), three even and seven odd (as with 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 and 587 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5), five even and 5 odd terms (as with 17939 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 3^5 + 3^5 + 7^5), seven even and 3 odd terms (as with 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5) or nine even terms and one odd term (as with 3413 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 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) = 41 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
a(2) = 103 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5.
a(3) = 227 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 283 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5.
a(5) = 587 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.

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

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

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, 10}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)

A000040 INTERSECTION A003355.

More terms from Alois P. Heinz, Aug 12 2015

A068537 Numbers which can be written as the sum of 2 like powers (x^n + y^n; n>1 & x,y>0).

Original entry on oeis.org

2, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 28, 29, 32, 33, 34, 35, 37, 40, 41, 45, 50, 52, 53, 54, 58, 61, 64, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 91, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 126, 128, 129, 130, 133, 136, 137, 145, 146

Offset: 1

Shawn Schafer (coolrelish(AT)hotmail.com), Mar 22 2002

nonn

			2 = 1^2 + 1^2; 5 = 1^2 + 2^2; 8 = 2^2 + 2^2; 9 = 1^3 + 2^3; 10 = 1^2 + 3^2; 13 = 2^2 + 3^2; 16 = 2^3 + 2^3; 17 = 1^2 + 4^2; ..... 33 = 1^5 + 2^5; etc...

Superset of A000404, A003325, A003336, A003347, A003358, A003369, A003380, A003391.

More terms from Michel Marcus, Aug 07 2013

More terms from Sean A. Irvine, Feb 21 2024

A291826 Numbers k such that k^5 is sum of 2 nonzero 6th powers.

Original entry on oeis.org

32, 2048, 23328, 131072, 500000, 1492992, 3764768, 8388608, 17006112, 32000000, 56689952, 95551488, 154457888, 240945152, 364500000, 536870912, 772402208, 1088391168, 1160290625, 1505468192, 2048000000, 2744515872, 3628156928, 4737148448, 6115295232

Offset: 1

XU Pingya, Sep 03 2017

nonn

If a^6 + b^6 = m, then (m^4*a)^6 + (m^4*b)^6 = m^25 = (m^5)^5 is 5th power. Therefore A003358(n)^5 is a term of this sequence for all n.
When k in this sequence, k*(n^6) (n >= 2) is also in this sequence.
If h = (i^6)*(j^6 + 1)^5 for (i >= 1 and j >= 1), then h is in this sequence. It appears that this equation generates all terms of the sequence. - Kieran Bhaskara, Aug 03 2019

			32^5 = 16^6 + 16^6, so 32 is in the sequence.
1160290625^5 = 17850625^6 + 35701250^6, so 1160290625 is in the sequence.

Cf. A000404, A004431, A003325, A050801, A003336, A003347, A003358.

lst={};Do[If[IntegerQ[(n^5-a^6)^(1/6)],AppendTo[lst,n]],{n,7*10^9},{a,(n^5/2)^(1/6)}]; lst

A291849 Numbers k such that k^3 is the sum of two nonzero 4th powers.

Original entry on oeis.org

8, 128, 648, 2048, 4913, 5000, 10368, 19208, 32768, 52488, 78608, 80000, 117128, 165888, 228488, 307328, 397953, 405000, 524288, 551368, 668168, 839808, 912673, 1042568, 1257728, 1280000, 1555848, 1874048, 2238728, 2654208, 3070625, 3125000, 3655808, 4251528

Offset: 1

XU Pingya, Sep 04 2017

nonn

If a^4 + b^4 = m, then (m^2 * a)^4 + (m^2 * b)^4 = m^9 = (m^3)^3 is a cube. Therefore A003336(n)^3 are terms of this sequence.

When k is in this sequence, k(n^4), for n > 1, is also in this sequence.

			8^3 = 2^9 = 4^4 + 4^4, so 8 is in the sequence.
4913^3 = 17^9 = 17^8 * (1 + 2^4) = 289^4 + 578^4, so 4913 is in the sequence.

Cf. A000404, A004431, A003325, A050801, A003336, A003347.

fourthPowerFlags = Union@Flatten@Table[a^4 + b^4 && GCD[a, b] == 1, {a, 4}, {b, a, 4}]; Take[Union@Flatten@Table[k^4 * fourthPowerFlags[[j]]^3, {k, 27}, {j, 6}], 34]

A291850 Numbers k such that k^2 is the sum of two positive 5th powers.

Original entry on oeis.org

8, 256, 1944, 6655, 8192, 25000, 35937, 62208, 134456, 212960, 262144, 344605, 453962, 472392, 692759, 800000, 1149984, 1288408, 1617165, 1990656, 2970344, 4302592, 6075000, 6814720, 8388608, 8732691, 11358856, 14526784, 15116544, 19808792, 20796875, 22168288

Offset: 1

XU Pingya, Sep 04 2017

nonn

If a^5 + b^5 = m, then (ma)^5 + (mb)^5 = m^6 = (m^3)^2 is square. Therefore A003347(n)^3 are terms of this sequence.

When k is in this sequence, k * (n^5) (n = 2, 3, ... ) is also in this sequence.

			8^2 = 2^5 + 2^5, so 8 is in the sequence.
6655^2 = 22^5 + 33^5, so 6655 is in the sequence.

Cf. A000404, A004431, A003325, A050801, A003336, A003347.

lst={};Do[If[IntegerQ[(n^2-a^5)^(1/5)],AppendTo[lst,n]],{n,9000},{a,(n^2/2)^(1/5)}]; lst

upto(n) = {
    my(res = List(), u = n^2, i5);
    for(i = 1, sqrtnint(u, 5),
        i5 = i^5;
        for(j = i, sqrtnint(u - i5, 5),
            c = i5 + j^5;
            if(issquare(c, &sc),
                listput(res, sc))));
    Set(res)} \\ David A. Corneth, Jun 17 2025

A291851 Numbers k such that k^3 is the sum of two positive 5th powers.

Original entry on oeis.org

4, 128, 972, 1089, 4096, 12500, 31104, 34848, 59536, 67228, 75625, 131072, 236196, 264627, 400000, 644204, 995328, 1050625, 1115136, 1485172, 1605289, 1905152, 2151296, 2420000, 3037500, 3403125, 4194304, 5679428, 7558272, 8468064, 9771876, 9904396, 9966649

Offset: 1

XU Pingya, Sep 04 2017

nonn

If a^5 + b^5 = m, then (ma)^5 + (mb)^5 = m^6 = (m^2)^3 is a cube. Therefore the square of each term of A003347 is a term of this sequence.

When k is in this sequence, k * (n^5), for n > 1, is also in this sequence.

			4^3 = 2^6 = 2^5 + 2^5, so 4 is in the sequence.
1089^3 = 33^5 + 66^5, so 1089 is in the sequence.

Cf. A000404, A004431, A003325, A050801, A003336, A003347.

lst={};Do[If[IntegerQ[(n^3-a^5)^(1/5)],AppendTo[lst,n]],{n,10^7},{a,(n^3/2)^(1/5)}]; lst

cp's OEIS Frontend

A123037 Prime sums of 8 positive 5th powers.

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A123038 Prime sums of 9 positive 5th powers.

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A123039 Prime sums of 11 positive 5th powers.

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A123040 Prime sums of 12 positive 5th powers.

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A123043 Prime sums of 10 positive 5th powers.

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A068537 Numbers which can be written as the sum of 2 like powers (x^n + y^n; n>1 & x,y>0).

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A291826 Numbers k such that k^5 is sum of 2 nonzero 6th powers.

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A291849 Numbers k such that k^3 is the sum of two nonzero 4th powers.

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