A126657 -id:A126657 - cp's OEIS frontend

A125516 Prime numbers that are the sum of three distinct positive squares.

29, 41, 53, 59, 61, 83, 89, 101, 107, 109, 113, 131, 137, 139, 149, 157, 173, 179, 181, 197, 211, 227, 229, 233, 241, 251, 257, 269, 277, 281, 283, 293, 307, 313, 317, 331, 337, 347, 349, 353, 373, 379, 389, 397, 401, 409, 419, 421, 433, 443, 449, 457, 461

Offset: 1

Tomas Xordan, Jan 21 2007

nonn

			29 = 2^2 + 3^2 + 4^2 = 4 + 9 + 16.
89 = 2^2 + 6^2 + 7^2 = 4 + 36 + 49; also 89 = 3^2 + 4^2 + 8^2 = 9 + 16 + 64.
353 = 2^2 + 5^2 + 18^2 = 4 + 25 + 324; also 353 = 4^2 + 9^2 + 16^2 = 16 + 81 + 256.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A088348, A126657, A126658.

Select[Total/@Subsets[Range[20]^2,{3}],PrimeQ]//Union (* Harvey P. Dale, Aug 12 2025 *)

{m=22;p=m^2;v=vector(m,x,x^2);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

A306214 Numbers that are the sum of fourth powers of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

98, 353, 707, 962, 1568, 2177, 2658, 3107, 4322, 4737, 5648, 7187, 7793, 7938, 9587, 11312, 12657, 13058, 15392, 15938, 17123, 19362, 20657, 23153, 23603, 25088, 28593, 30963, 31202, 32738, 34832, 35747, 40962, 42528, 45233, 45377, 49712, 49763, 54722, 57153, 57267, 61250, 63938, 67667, 69152

Offset: 1

Antonio Roldán, Jan 29 2019

nonn

The remainder of a(n) divided by 16 is less than 5. - Jinyuan Wang, Feb 03 2019

			353 = 2^4 + 3^4 + 4^4, with 3 - 2 = 4 - 3 = 1;
7187 = 1^4 + 5^4 + 9^4, with 5 - 1 = 9 - 5 = 4.

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

Cf. A000583, A126657, A133531, A190176, A306212, A306213.

N:= 10^5: # for all terms <= N
Res:= NULL:
for a from 1 to floor((N/3)^(1/4)) do
  for d from 1 do
    v:= a^4 + (a+d)^4 + (a+2*d)^4;
    if v > N then break fi;
    Res:= Res, v
  od
od:
sort(convert({Res},list)); # Robert Israel, Feb 17 2019

for(n=1, 70000, k=(n/3)^(1/4); a=2; v=0; while(a<=k&&v==0, d=sqrt(sqrt(2*n+30*a^4)/2-3*a^2); if(d==truncate(d)&&d>=1&&d<=a-1, v=1; print1(n,", ")); a+=1))

A126112 Prime numbers p such that p^4 + (p-1)^4 + (p+1)^4 is a prime number.

Original entry on oeis.org

3, 7, 11, 29, 31, 53, 59, 83, 109, 127, 283, 349, 461, 521, 599, 643, 683, 787, 809, 829, 907, 911, 937, 983, 1093, 1117, 1201, 1289, 1301, 1487, 1523, 1613, 1721, 1877, 2017, 2153, 2267, 2281, 2423, 2521, 2579, 2657, 2677, 2699, 2731, 2741, 2797, 2887, 2969

Offset: 1

Tomas Xordan, Mar 05 2007

nonn

			(3-1)^4 + 3^4 + (3+1)^4 = 2^4 + 3^4 + 4^4 = 16 + 81 + 256 = 353 is prime, hence 3 is a term.
(11-1)^4 + 11^4 + (11+1)^4 = 10^4 + 11^4 + 12^4 = 10000 + 14641 + 20736 = 45377 is prime, hence 11 is a term.

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

Cf. A126657, A126769, A126113.

f[n_]:=PrimeQ[(n-1)^4+n^4+(n+1)^4];lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Select[Prime[Range[500]],PrimeQ[Total[(#+{-1,0,1})^4]]&] (* Harvey P. Dale, Dec 07 2012 *)

forprime(p=2, 3000, if(isprime(q=(p-1)^4+p^4+(p+1)^4), print1(p, ","))) /* Klaus Brockhaus, Mar 09 2007 */

Edited, corrected and extended by Klaus Brockhaus, Mar 09 2007

A126113 Primes p^4 + (p-1)^4 + (p + 1)^4 arising from A126112.

Original entry on oeis.org

353, 7793, 45377, 2131937, 2782097, 23705153, 36393857, 142457633, 423617057, 780627473, 19243704833, 44507912417, 135498076577, 221043906737, 386218778417, 512825188193, 652841559233, 1150861306913, 1285043991857

Offset: 1

Tomas Xordan, Mar 05 2007

nonn

			A126112(1) = 3 and (3-1)^4+3^4+(3+1)^4 = 2^4+3^4+4^4 = 16+81+256 = 353 is prime, hence a(1) = 353.
A126112(3) = 11 and (11-1)^4+11^4+(11+1)^4 = 10^4+11^4+12^4 = 10000 + 14641 + 20736 = 45377 is prime, hence a(3) = 45377.

Cf. A126657, A126769, A126112.

{forprime(p=2,810,if(isprime(q=(p-1)^4+p^4+(p+1)^4),print1(q,",")))} /* Klaus Brockhaus, Mar 27 2007 */

Edited and corrected by Klaus Brockhaus, Mar 27 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

A140834 Primes that are the sum of at most four nonzero 4th powers.

Original entry on oeis.org

2, 3, 17, 19, 83, 97, 113, 163, 179, 257, 337, 353, 419, 499, 593, 641, 643, 673, 769, 787, 881, 883, 1153, 1297, 1409, 1459, 1553, 1889, 2003, 2083, 2131, 2417, 2579, 2593, 2609, 2657, 2659, 2689, 2819, 3169, 3217, 3697, 3779, 3889, 3907, 4099, 4129, 4177

Offset: 1

Jonathan Vos Post, Jul 18 2008

This sequence was checked by T. D. Noe, who had supplied the b-list for A004833. A037896 is a subset of {Primes that are the sum of at exactly 2 nonzero 4th powers}, itself a subset of A002645 Quartan primes: primes of the form x^4 + y^4, x>0, y>0.

Cf. A000040, A000583, A002645, A003294, A004833, A037896, A085318, A122540, A126657, A133740.

A000040 INTERSECTION A004833. {A133740 = Primes that are the sum of at exactly 4 nonzero 4th powers} UNION {A085318 = Primes that are the sum of at exactly 3 nonzero 4th powers} UNION {A002645 = Primes that are the sum of at exactly 2 nonzero 4th powers}.

Missing term 353 inserted by Georg Fischer, May 11 2024

cp's OEIS Frontend

A125516 Prime numbers that are the sum of three distinct positive squares.

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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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A306214 Numbers that are the sum of fourth powers of three distinct positive integers in arithmetic progression.

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A126112 Prime numbers p such that p^4 + (p-1)^4 + (p+1)^4 is a prime number.

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A126113 Primes p^4 + (p-1)^4 + (p + 1)^4 arising from A126112.

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

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Maple

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A140834 Primes that are the sum of at most four nonzero 4th powers.

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