A125516 -id:A125516 - cp's OEIS frontend

A126657 Prime numbers that are the sum of three distinct positive fourth powers.

353, 1553, 5393, 6833, 7187, 7793, 7873, 8963, 9043, 9587, 10337, 11953, 13697, 14177, 14723, 16193, 17123, 20753, 21283, 21377, 21617, 23603, 25457, 28643, 29873, 30113, 30817, 31393, 35393, 35747, 39857, 43283, 45233, 45377, 46273

Offset: 1

Tomas Xordan, Feb 09 2007

nonn

			1553= 1^4 + 4^4 + 6^4 = 1 + 256 + 1296.
6833 = 2^4 + 4^4 + 9^4 = 16 + 256 + 6561.
21377 = 2^4 + 5^4 + 12^4 = 16 + 625 + 20736.
35747 = 5^4 + 9^4 + 13^4 = 625 + 6561 + 28561.

Cf. A125516, A126658.

Union[Select[Total/@Subsets[Range[20]^4,{3}],PrimeQ]] (* Harvey P. Dale, May 08 2012 *)

{m=15;p=m^4;v=vector(m,x,x^4);w=[];for(i=1,m-2,for(j=i+1,m-1, for(k=j+1,m,if((n=v[i]+v[j]+v[k])

Edited, corrected and extended by Klaus Brockhaus, Feb 11 2007

A126658 Prime numbers that are the sum of three distinct positive eighth powers.

Original entry on oeis.org

72353, 1745153, 7444673, 44726593, 49202147, 61503553, 100006817, 100072097, 101686177, 107444417, 143046977, 214756067, 257412163, 430372577, 431661313, 435812033, 447149537, 452523713, 489805633, 530372321, 744340577, 834187553

Offset: 1

Tomas Xordan, Feb 09 2007

nonn

These are also the sum of three squares and the sum of three fourth powers: 7444673 = 16^2 + 1296^2 + 2401^2 = 4^4 + 36^4 + 49^4 = 256 + 1679616 + 5764801.

			72353 = 2^8 + 3^8 + 4^8 = 256 + 6561 + 65536.
7444673 = 2^8 + 6^8 + 7^8 = 256 + 1679616 + 5764801.
49202147 = 5^8 + 7^8 + 9^8 = 390625 + 5764801 + 43046721.

Cf. A125516, A126657.

{m=14;p=m^8;v=vector(m,x,x^8);w=[];for(i=1,m-2,for(j=i+1,m-1, for(k=j+1,m,if((n=v[i]+v[j]+v[k])

Edited, corrected and extended by Klaus Brockhaus, Feb 11 2007

A126708 Prime numbers that are the sum of the cubes of three distinct primes with the same final digit.

Original entry on oeis.org

93871, 100043, 159389, 161071, 236627, 240551, 297233, 325693, 409499, 456623, 468551, 524287, 550061, 583981, 614683, 617401, 653491, 705277, 722807, 800171, 968239, 1016839, 1040311, 1129013, 1172261, 1276039, 1317259, 1326277, 1379519

Offset: 1

Tomas Xordan, Feb 11 2007

			93871 = 13^3 + 23^3 + 43^3 = 2197 + 12167 + 79507 is prime and 13, 23, 43 are primes with the same final digit, hence 93871 is a term.
617401 = 43^3 + 53^3 + 73^3 = 79507 + 148877 + 389017 is prime and 43, 53, 73 are primes with the same final digit, hence 617401 is a term.
14391 = 3^3 + 13^3 + 23^3 = 27 + 2197 + 12167 is not prime; although 3, 13, 23 are primes with the same final digit, 14391 is not in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A125516, A126657, A126658.

{m=116; p=m^3; w=[]; forprime(i=1, m-2, r=i%10; forprime(j=i+1, m-1, forprime(k=j+1, m, if(j%10==r&&k%10==r&&(n=i^3+j^3+k^3)

Edited, corrected and extended by Klaus Brockhaus, Feb 16 2007

A126704 Prime numbers that are the sum of three distinct positive sixth powers.

Original entry on oeis.org

4889, 50753, 51481, 66377, 262937, 308801, 797681, 840241, 1000793, 1046657, 1772291, 2303003, 2986777, 3032641, 3107729, 3365777, 4757609, 4804201, 5135609, 7530329, 7534361, 8061041, 8065073, 10516249, 12394721, 14638753

Offset: 1

Tomas Xordan, Feb 11 2007

nonn

			4889 = 2^6 + 3^6 + 4^6 = 64 + 729 + 4096.
66377 = 4^6 + 5^6 + 6^6 = 4096 + 15625 + 46656.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A125516, A126657, A126658.

N:= 10^10; # to find all terms <= N
A := {}:
for a from 1 to iroot(N,6) do
  for b from 1 to a-1 while a^6 + b^6 < N do
    for c from (a+b) mod 2 + 1 to b-1 by 2 do
      r:= a^6 + b^6 + c^6;
      if r > N then break fi;
      if isprime(r) then A:= A union {r} fi;
od od od:
sort(convert(A,list)); # Robert Israel, Dec 15 2015

Union[Select[Total/@Subsets[Range[20]^6,{3}],PrimeQ]] (* Harvey P. Dale, Apr 20 2013 *)

{m=16; p=m^6; w=[]; for(i=1,m-2,for(j=i+1, m-1, for(k=j+1, m, if((n=i^6+j^6+k^6)Klaus Brockhaus, Feb 16 2007 */

Edited, corrected and extended by Klaus Brockhaus, Feb 16 2007

A242675 Smallest prime with exactly n representations as sum of 3 distinct positive squares.

Original entry on oeis.org

2, 29, 89, 101, 281, 269, 641, 461, 701, 761, 1049, 941, 1109, 1601, 1361, 2621, 2309, 1889, 2441, 2141, 2609, 3929, 3701, 3461, 3449, 5849, 4241, 4289, 5081, 8669, 7589, 5381, 6569, 9941, 8861, 7229, 7829, 8501, 8069, 8609, 9749, 10601

Offset: 0

Zak Seidov, May 20 2014

2 cannot be represented as the sum of 3 distinct positive squares hence a(0)=2 (and offset is 0).

			29 = 2^2 + 3^2 + 4^2 and this is the only such representation.
89 = 2^2 + 6^2 + 7^2 = 3^2 + 4^2 + 8^2 and these are the only such representations.
101 = 1^2 + 6^2 + 8^2 = 2^2 + 4^2 + 9^2 = 4^2 + 6^2 + 7^2 and these are the only such representations.

Zak Seidov, Table of n, a(n) for n = 0..1000

Cf. A004432, A125516.

A263723 Number of representations of the prime P = A182479(n) as P = p^2 + q^2 + r^2, where p < q < r are also primes.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 4, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

nonn

According to Sierpinski and Schinzel (1988), it is easy to prove that the smallest of p, q, r is always p = 3, and under Schinzel's hypothesis H the sequence is infinite.

			A182479(1) = 83 = 3^2 + 5^2 + 7^2 and A182479(2) = 179 = 3^2 + 7^2 + 11^2 are the only ways to write 83 and 179 as sums of squares of 3 distinct primes, so a(1) = 1 and a(2) = 1.
A182479(5) = 419 = 3^2 + 7^2 + 19^2 = 3^2 + 11^2 + 17^2 are the only such representations of 419, so a(5) = 2.

W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988; see pp. 220-221.

Wikipedia, Schinzel's Hypothesis H

Cf. A085317, A125516, A182479.

lst = {}; r = 7; While[r < 132, q = 5; While[q < r, P = 9 + q^2 + r^2; If[PrimeQ@P, AppendTo[lst, P]];
  q = NextPrime@q]; r = NextPrime@r]; Take[Transpose[Tally@Sort@lst][[2]], 105]

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A126657 Prime numbers that are the sum of three distinct positive fourth powers.

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A126658 Prime numbers that are the sum of three distinct positive eighth powers.

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A126708 Prime numbers that are the sum of the cubes of three distinct primes with the same final digit.

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A126704 Prime numbers that are the sum of three distinct positive sixth powers.

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A242675 Smallest prime with exactly n representations as sum of 3 distinct positive squares.

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A263723 Number of representations of the prime P = A182479(n) as P = p^2 + q^2 + r^2, where p < q < r are also primes.

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