A114923 Primes p such that there exist three primes q, r and s with p^3=q^3+r^3+s^3.

709, 1033, 2767, 2791, 2917, 3727, 3769, 5647, 5657, 5737, 7039, 7321, 8089, 8291, 8387, 9433, 9473, 9851, 12073, 12343, 13417, 14083, 14561, 14723, 14831, 14969, 15313, 18127, 19841, 25033, 28081, 28477, 29153, 29179, 32771, 33161, 33199, 33377, 34337, 36713

Offset: 1

Author

Carlos Rivera and Farideh Firoozbakht, Jan 09 2006

Keywords

• nonn

Comments

The sets of three primes corresponding to the first seven terms of the sequence are respectively {193,461,631}, {599,691,823}, {103,2179,2213}, {769,1879,2447}, {31,1951,2591}, {1399,1667,3541} and {11,1783,3631}. - Robert G. Wilson v, Jan 09 2006

The sets of three primes corresponding to the next eight terms of the sequence are respectively {2251, 3121, 5171}, {1487, 2731, 5399}, {839, 3691, 5167}, {2099, 2377, 6883}, {3163, 5443, 5843}, {1621, 6323, 6481}, {2357, 4999, 7559} and {1621, 5297, 7589}. - Robert G. Wilson v, Jan 09 2006

The indices of the primes: 127,174,403,406,422,520,525,742,745,754,905,933,1017,1040,1050, ..., . - Robert G. Wilson v, Jan 09 2006

The sets of three primes corresponding to the terms 12073, 12343, 13417, 14083, 14561, 14723, 14831, 14969, 15313, 18127, 19841 and 25033 are respectively {4007, 4327, 11731}, {373, 9209, 10321}, {5099, 7561, 12277}, {4639, 7129, 13259}, {1997, 8599, 13469}, {3881, 6427, 14207}, {6257, 9439, 12959}, {2239, 5189, 14741}, {2269, 2969, 15259}, {2129, 5227, 17971}, {3931, 15263, 16127} and {4093, 19391, 20269}. The indices of the primes: 127, 174, 403, 406, 422, 520, 525, 742, 745, 754, 905, 933, 1017, 1040, 1050, 1168, 1174, 1215, 1446, 1474, 1591, 1661, 1707, 1723, 1738, 1753, 1789, 2077, 2244, 2765. - Farideh Firoozbakht, Jan 27 2006

Examples

			The prime number 3769 is in the sequence because we have 3769^3=11^3+1783^3+3631^3 and three numbers 11, 1783 and 3631 are primes.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..776 (first 121 terms from Chai Wah Wu)
• G. L. Honaker, Jr. and Chris Caldwell, Prime Pages.
• Carlos Rivera, P^3=a^3+b^3+c^3, {P, a, b, c} = primes, Puzzle 48, The Prime Puzzles & Problems Connection.

Crossrefs

Subset of A023042.

Programs

Maple

N:= 20000: # to get all terms <= N
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
P2:= {seq(seq(Primes[i]^3 + Primes[j]^3, j=1..i),i=1..nops(Primes))}:
Q:= convert(map(t->-t^3,Primes),set):
filter:= p -> P2 intersect map(`+`,Q,p^3) <> {}:
select(filter, Primes); # Robert Israel, Jan 11 2016

Mathematica

t = {}; Do[ If[p = (Prime[q]^3 + Prime[r]^3 + Prime[s]^3)^(1/3); PrimeQ[p], AppendTo[t, p]; Print[{p, Prime[s], Prime[r], Prime[q]}]], {q, 3, 1059}, {r, q-1}, {s, r-1}]; t (* Robert G. Wilson v, Jan 09 2006 *)

PARI

is(p)=my(p3=p^3,a3,A,c);if(isprimepower(p3-16)==3, return(1)); forprime(a=sqrtnint(p3\3,3),sqrtnint(p3-54,3), a3=a^3; A=p3-a3; forprime(b=3,min(sqrtnint(A,3),a), if(ispower(A-b^3,3,&c) && isprime(c), return(isprime(p))))) \\ Charles R Greathouse IV, Nov 24 2017

Extensions

a(8)-a(18) from Robert G. Wilson v, Jan 09 2006

a(19)-a(30) from Farideh Firoozbakht, Jan 27 2006

a(31)-a(40) from Chai Wah Wu, Jan 10 2016