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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.
4
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
1,1
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
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.
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
Subset of A023042.
Sequence in context: A216402 A216315 A059312 * A057849 A199091 A058324
KEYWORD
nonn
AUTHOR
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
STATUS
approved