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

%I A114923 #29 Apr 03 2023 10:36:11
%S A114923 709,1033,2767,2791,2917,3727,3769,5647,5657,5737,7039,7321,8089,8291,
%T A114923 8387,9433,9473,9851,12073,12343,13417,14083,14561,14723,14831,14969,
%U A114923 15313,18127,19841,25033,28081,28477,29153,29179,32771,33161,33199,33377,34337,36713
%N A114923 Primes p such that there exist three primes q, r and s with p^3=q^3+r^3+s^3.
%C A114923 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
%C A114923 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
%C A114923 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
%C A114923 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
%H A114923 Charles R Greathouse IV, <a href="/A114923/b114923.txt">Table of n, a(n) for n = 1..776</a> (first 121 terms from Chai Wah Wu)
%H A114923 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/page.php/709.html">Prime Pages</a>.
%H A114923 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_048.htm">P^3=a^3+b^3+c^3, {P, a, b, c} = primes</a>, Puzzle 48, The Prime Puzzles & Problems Connection.
%e A114923 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.
%p A114923 N:= 20000: # to get all terms <= N
%p A114923 Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
%p A114923 P2:= {seq(seq(Primes[i]^3 + Primes[j]^3, j=1..i),i=1..nops(Primes))}:
%p A114923 Q:= convert(map(t->-t^3,Primes),set):
%p A114923 filter:= p -> P2 intersect map(`+`,Q,p^3) <> {}:
%p A114923 select(filter, Primes); # _Robert Israel_, Jan 11 2016
%t A114923 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 *)
%o A114923 (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
%Y A114923 Subset of A023042.
%K A114923 nonn
%O A114923 1,1
%A A114923 _Carlos Rivera_ and _Farideh Firoozbakht_, Jan 09 2006
%E A114923 a(8)-a(18) from _Robert G. Wilson v_, Jan 09 2006
%E A114923 a(19)-a(30) from _Farideh Firoozbakht_, Jan 27 2006
%E A114923 a(31)-a(40) from _Chai Wah Wu_, Jan 10 2016