A130833 Sums of two or more distinct 4th powers of primes.

97, 641, 706, 722, 2417, 2482, 2498, 3026, 3042, 3107, 3123, 14657, 14722, 14738, 15266, 15282, 15347, 15363, 17042, 17058, 17123, 17139, 17667, 17683, 17748, 17764, 28577, 28642, 28658, 29186, 29202, 29267, 29283, 30962, 30978, 31043, 31059, 31587, 31603

Offset: 1

Author

Jonathan Vos Post, Jul 21 2011

Keywords

• nonn
• easy

Comments

This is to cubes and A030078 as A192926 is to 4th powers and A030514. The subsequence of primes which are sums of two or more distinct 4th powers of primes begins 97, 641, 2417 (A193411).

The first term that arises in more than one way is 6539044 = 11^4 + 23^4 + 41^4 + 43^4 = 13^4 + 29^4 + 31^4 + 47^4. - Robert Israel, Apr 27 2020

Examples

			a(1) = 97 = 2^4 + 3^4.
a(2) = 641 = 2^4 + 5^4.
a(3) = 706 = 3^4 + 5^4.
a(4) = 722 = 2^4 + 3^4 + 5^4.
		

Links

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

Crossrefs

Cf. A000040, A000583, A030514, A192926, A193411.

Programs

Maple

N:= 40000: # for all terms <= N
S1:= {}:
S2:= {}:
p:= 1:
do
  p:= nextprime(p);
  if p^4 > N then break fi;
  s:= p^4;
  S2:= S2 union select(`<=`,map(`+`,S1 union S2, s), N);
  S1:= S1 union {s};
od:
sort(convert(S2,list)); # Robert Israel, Apr 27 2020

Mathematica

nn=6; t = Sort@ Flatten@ Table[ n^4, {n, Prime@ Range@ nn}]; Select[Sort[
Plus @@@ Subsets[t, {2, nn}]], # < Prime[nn-1]^4 + Prime[nn]^4 &] (* Robert G. Wilson v, Jul 22 2011 *)

Formula

{A030078(i) + A030078(j) for i not equal to j} UNION {A030078(i) + A030078(j) + A030078(k) for i not equal to j not equal to k} UNION {A030078(i) + A030078(j) + A030078(k) + A030078(L) for i not equal to j not equal to k not equal to L}...