A003336 -id:A003336 - cp's OEIS frontend

A123035 Prime sums of 6 positive 5th powers.

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

A123037 Prime sums of 8 positive 5th powers.

Original entry on oeis.org

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

A182277 Quartan semiprimes: semiprimes of the form x^4 + y^4, x>0, y>0.

Original entry on oeis.org

82, 626, 706, 1921, 2402, 4097, 6497, 6817, 7186, 8962, 10001, 10081, 14642, 17042, 18737, 20737, 21202, 21361, 23137, 24641, 28562, 28642, 29186, 29857, 35377, 38417, 38497, 43202, 44977, 50641, 53026, 53057, 65266, 67937, 72097, 83522, 83602, 84146, 84817, 85922

Offset: 1

Jonathan Vos Post, Apr 22 2012

nonn

This is to A002645 as A001358 semiprimes is to A000040 primes.

			a(1) = 3^4 + 1^4 = 82 = 2 * 41.

George Greaves, On the representation of a number as a sum of two fourth powers, MATHEMATISCHE ZEITSCHRIFT, Volume 94, Number 3 (1966), 223-234, DOI: 10.1007/BF01111351.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A003336 Numbers that are the sum of 2 nonzero 4th powers, A002645 Quartan primes: primes of the form x^4 + y^4, x>0, y>0.

issemi(n)=bigomega(n)==2
list(lim)=my(v=List(),t);for(x=1,(lim+.5)^(1/4),for(y=1,min(x,(lim-x^4 + .5)^(1/4)),if(issemi(t=x^4+y^4),listput(v,t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Apr 22 2012

A001358 INTERSECTION A003336.

a(12)-a(40) from Charles R Greathouse IV, Apr 22 2012

A209431 Numbers n such that x^4 + y^4 = n * z^4 is solvable in nonzero integers x,y,z with z > 1 and gcd(x,y,n) = 1.

Original entry on oeis.org

5906, 469297, 926977, 952577, 1127857, 1298257, 1347361, 1647377, 2455361, 3342817, 4928977, 5268706, 5519537, 8588161, 8879537, 9339361, 9391537, 9846017, 11414017, 14543026, 15547297, 16502722, 16657217, 16672322, 16830017, 19730162, 23672002, 25030097, 27681937, 27979762

Offset: 1

Jean-François Alcover, Mar 09 2012

nonn

Values of z (1, 17, 41, 73, 89, ...) are elements of sequence A004625 (divisible only by primes congruent to 1 mod 8). The first composite z is 697 = 17*41: 41^4 + 822091^4 = 1935300738962*697^4.

Proof (after Ms. Adina Calvo) that values of z are divisible only by primes congruent to 1 mod 8: Let {x,y,z} be a nontrivial solution and p an odd prime divisor of z. Reducing the equation mod p, one gets in Z/pZ: x^4 + y^4 = 0 mod p. Hence (x*y^-1)^4 = -1, then x*y^-1 is an order-8 element of the multiplicative group (Z/pZ)*, which has p-1 elements. Therefore p is congruent to 1 mod 8.

			5906 is in the sequence because a^4 + b^4 = 5906*c^4 has the solution (a,b,c) = (25,149,17).

A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
Steven R. Finch, On a generalized Fermat-Wiles equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A060387, A003336, A020897.

BiquadraticFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 4]} & /@ FactorInteger[n]); max = 10000; Sort[ Reap[Do[nz4 = x^4 + y^4; z4 = nz4/BiquadraticFreePart[nz4]; z = z4^(1/4); n = nz4/z4; If[z4 > 1 && IntegerQ[z] && GCD[x, y, z] == 1, Print[{n, x, y, z}]; Sow[n]], {x, 1, max}, {y, x, max}]][[2, 1]]]

Numbers in A060387 but not in A003336.

Definition corrected by Hugo Pfoertner, Nov 08 2016

A282948 Numbers n such that (u^4 + v^4)/2 = x^4 + y^4 = n has a solution in positive integers u,v,x,y.

Original entry on oeis.org

162401, 2598416, 13154481, 41574656, 101500625, 210471696, 389924801, 665194496, 1065512961, 1624010000, 2377713041, 3367547136, 4638334961, 6238796816, 8221550625, 10643111936, 13563893921, 17048207376, 21164260721, 25984160000, 31583908881, 38043408656

Offset: 1

Altug Alkan and Thomas Ordowski, Feb 25 2017

nonn

All terms are composite.
If n is in this sequence, then n*k^4 with k > 0 is in this sequence.
Numbers n such that n and 2*n are both in A003336. - Michel Marcus, Feb 25 2017
The first term which is not a multiple of a(1) is a(84) = 8051889328801. - Giovanni Resta, Feb 25 2017
Based on Giovanni Resta's b-file, the squarefree terms are 162401, 8051889328801, 9305528350081, 16778006844241, .... - Altug Alkan, Feb 26 2017
Izadi & Nabardi construct a collection of elliptic curves of rank >= 5 using (essentially) terms of this sequence. - Charles R Greathouse IV, Jul 13 2024

			(19^4 + 21^4)/2 = 7^4 + 20^4 = 162401.

Giovanni Resta, Table of n, a(n) for n = 1..513 (terms < 10^16)
Farzali Izadi and Kamran Nabardi, A Family of Elliptic Curves With Rank >= 5, arXiv preprint (2015). arXiv:1501.03809 [math.NT]

Cf. A003336, A191345.

isA003336(n) = for(k=1, sqrtnint(n\2, 4), ispower(n-k^4, 4) && return(1));
is(n) = isA003336(n) && isA003336(2*n);

T=thueinit('x^4+1, 1);
has(n)=#thue(T, n)>0 && !issquare(n)
list(lim)=my(v=List(),x4,t); for(x=1,sqrtnint(lim\=1,4), x4=x^4; for(y=1,min(sqrtnint(lim-x4,4),x), t=x4+y^4; if(has(2*t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 26 2017

a(10)-a(22) from Giovanni Resta, Feb 25 2017

cp's OEIS Frontend

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