A134657 Numbers of the form p^2 + q^3 + r^4 with p, q and r primes.

28, 33, 47, 49, 52, 68, 73, 92, 93, 98, 112, 114, 117, 133, 138, 145, 150, 157, 164, 166, 190, 193, 210, 212, 215, 229, 231, 255, 258, 262, 277, 310, 313, 327, 332, 363, 368, 375, 378, 384, 385, 397, 404, 408, 428, 430, 433, 449, 450, 469, 473, 480, 495

Offset

1,1

Comments

The primes p, q, and r are not necessarily distinct. - Jon E. Schoenfield, Sep 24 2018

Links

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

Project Euler, Problem 87: Prime power triples.

Yuhui Liu, On sums of unequal powers of primes and powers of 2, arXiv preprint arXiv:2401.01355 [math.NT], 2023.

Formula

Liu proves that a(n) < 37.64071*n for large enough n. "Large enough" is probably in the neighborhood of 10^30 to 10^40. - Charles R Greathouse IV, Jan 12 2026

Example

a(1) = 28 = 2^2 + 2^3 + 2^4 is the smallest sum of a prime squared, a prime cubed and the 4th power of a prime.
a(2) = 33 = 3^2 + 2^3 + 2^4 is the next number of that form.

Maple

N:= 1000: # for terms <= N
P:= select(isprime, [2, seq(i, i=3..floor(sqrt(N)), 2)]):
S:= {}:
for i from 1 to nops(P) do
  t:= P[i]^2;
  for j from 1 do
    u:= t + P[j]^3; if u > N then break fi;
    for k from 1 do
      v:= u + P[k]^4; if v > N then break fi;
      S:= S union {v}
od od od:
sort(convert(S, list)); # Robert Israel, Mar 02 2026

Mathematica

Take[Union[Total[#^{2, 3, 4}]&/@Tuples[Prime[Range[10]], 3]], 60] (* Harvey P. Dale, Mar 02 2013 *)

Prog

(PARI) is_p87(n, t, tt)=forprime(p=2, sqrtn(n, 4), t=n-p^4; forprime(q=2, sqrtn(t, 3), issquare(t-q^3, &tt) || next; isprime(tt) && return(1)))
print_p87(Nmax=999)=for(n=1, Nmax, is_p87(n) && print1(n", "))
(PARI) list(lim)=my(v=List()); if(lim<28, return([])); forprime(p=2, sqrtnint(-12+lim\=1, 4), my(p4=p^4); forprime(q=2, sqrtnint(lim-p4-4, 3), my(t=p4+q^3); forprime(r=2, sqrtint(lim-t), listput(v, t+r^2)))); Set(v) \\ Charles R Greathouse IV, Jan 12 2026

Crossrefs

Cf. A045701.

Sequence in context: A090637 A261113 A280649 * A261106 A375819 A138687

Adjacent sequences: A134654 A134655 A134656 * A134658 A134659 A134660

Keyword

easy,nonn

Author

M. F. Hasler, Jan 25 2008

Status

approved