A123597 -id:A123597 - cp's OEIS frontend

A182479 Primes of the form p^2 + q^2 + r^2, where p,q,r are distinct primes.

83, 179, 227, 347, 419, 467, 491, 563, 587, 659, 827, 971, 1019, 1091, 1259, 1427, 1499, 1667, 1811, 1907, 1979, 2027, 2243, 2267, 2339, 2531, 2579, 2699, 2819, 2843, 2939, 3347, 3539, 3659, 3779, 3851, 4019, 4091, 4259, 4451, 4523, 4547, 4691, 4787, 5099

Offset: 1

Alex Ratushnyak, May 01 2012

nonn

All terms are congruent to 5 modulo 6. Smallest of primes p, q, r is always 3. - Zak Seidov, Jun 03 2014

The number of such representations of a prime of that form is A263723. - Jonathan Sondow and Robert G. Wilson v, Nov 02 2015

			5099 = 3^2 + 7^2 + 71^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A123592, A123597, A137365, A263723.

Cf. A137364 (the same with repetitions). - Zak Seidov, Jun 03 2014

mx = 20; ps = Prime[Range[2, mx + 1]]; t = Table[ps[[i]]^2 + ps[[j]]^2 + ps[[k]]^2, {i, mx}, {j, i + 1, mx}, {k, j + 1, mx}]; Select[Union[Flatten[t]], # <= 34 + ps[[-1]]^2 && PrimeQ[#] &] (* T. D. Noe, May 01 2012 *)

list(lim)=my(v=List(),t);lim\=1;forprime(p=7,sqrt(lim), forprime(q=5,min(sqrtint(lim-p^2-9),p-1), t=p^2+q^2;forprime(r=3,min(sqrtint(lim-t),q-1), if(isprime(t+r^2), listput(v,t+r^2))))); vecsort(Vec(v),,8)
\\ Charles R Greathouse IV, May 01 2012

A353249 Primes that are the sum of the cubes of four primes, not necessarily distinct.

Original entry on oeis.org

89, 149, 367, 383, 503, 601, 1709, 2221, 2357, 4001, 4937, 5171, 6599, 6883, 7019, 7237, 7243, 7583, 9091, 10177, 11261, 11807, 14747, 15923, 16693, 17431, 24413, 24767, 25673, 26539, 27059, 30169, 32587, 34739, 43517, 48731, 51031, 51347, 53201, 53323, 53699, 54133, 59617

Offset: 1

Michel Marcus, Apr 08 2022

nonn

			89 is a term because 2^3 + 3^3 + 3^3 + 3^3 = 89.
15923 is a term because 2^3 + 13^3 + 19^3 + 19^3 = 15923.

Zhichun Zhai, Problems related to Waring-Goldbach problem involving cubes of primes, arXiv:2201.07346 [math.GM], 2022. See Table 1 p. 3 but some terms are missing.

Primes in A346917.

Cf. A123597.

q:= proc(n, t) option remember; `if`(n=0, is(t=0), t>0 and
      ormap(p-> isprime(p) and q(n-p^3, t-1), [$2..iroot(n, 3)]))
    end:
select(x-> isprime(x) and q(x, 4), [$1..60000])[];  # Alois P. Heinz, Apr 08 2022

seq[max_] := Module[{s = Select[Range[Floor@Surd[max, 3]], PrimeQ]}, Select[Union[Plus @@@ (Tuples[s, 4]^3)], # <= max && PrimeQ[#] &]]; seq[60000] (* Amiram Eldar, Apr 08 2022 *)

isok(p) = {if (isprime(p) && (p > 24), my(P=primes(primepi(sqrtn(p-24, 3)+1))); for (i=1, #P, for (j=i, #P, for (k=j, #P, for (n=k, #P, if (P[i]^3 + P[j]^3 + P[k]^3 + P[n]^3 == p, return (1)););););););}

cp's OEIS Frontend

A182479 Primes of the form p^2 + q^2 + r^2, where p,q,r are distinct primes.

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