A318530 - OEIS

%I #19 Mar 09 2026 09:17:42

%S 145,210,637,754,2317,2530,2917,5218,5437,5890,6447,6997,7469,7653,

%T 7738,8650,9333,11818,12417,12796,14770,15178,15197,15295,15513,16349,

%U 16501,17367,18389,19709

%N Numbers that can be written in more than one way as p^2 + q^3 + r^4 with p, q and r primes.

%C Theorem: a number n can be written in at most n^(11/30 + o(1)) ways as p^2 + q^3 + r^4. Proof: a result of Pila (see Links) shows that there are at most x^(1/3 + o(1)) solutions to a cubic curve, hence fix r and there are at most n^(1/6 + o(1)) solutions with p,q <= sqrt(n); multiplying by the n^(1/5) choices for r yields the result. - _Charles R Greathouse IV_, Mar 09 2026

%H Robert Israel, <a href="/A318530/b318530.txt">Table of n, a(n) for n = 1..10000</a>

%H J. Pila, <a href="https://www.numdam.org/article/AST_1995__228__183_0.pdf">Density of integral and rational points on varieties</a>, Astérisque, tome 228 (1995), pp. 183-187.

%F n << a(n) << n^(60/43 + o(1)). The exponent is 1/(1/2+1/3+1/4-11/30). - _Charles R Greathouse IV_, Mar 09 2026

%e a(1) = 145 = 2^2 + 5^3 + 2^4 = 11^2 + 2^3 + 2^4 .

%e The first term which can be written in three different ways is 17367 = 23^2 + 13^3 + 11^4 = 113^2 + 13^3 + 7^4 = 131^2 + 5^3 + 3^4 .

%p N:= 10^5: # to get terms <= N

%p P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)))]):

%p Psq:= map(`^`,P,2):

%p P3:= select(`<=`,map(`^`,P,3),N):

%p P4:= select(`<=`,map(`^`,P,4),N):

%p V:= Vector(N):

%p for a in Psq do for b in P3 do for c in P4 do

%p    s:= a+b+c;

%p    if s <= N then V[s]:= V[s]+1 fi

%p od od od:

%p select(t -> V[t]>=2, [$1..N]); # _Robert Israel_, Jan 30 2019

%Y Subsequence of A134657.

%K nonn

%O 1,1

%A _Philippe Guglielmetti_, Aug 28 2018