A276415 - OEIS

%I #26 Oct 06 2016 02:44:23

%S 0,1,3,3,2,2,2,2,1,1,3,3,3,3,2,1,1,3,5,4,3,2,2,3,2,2,3,2,4,5,5,4,3,2,

%T 3,1,3,4,4,3,3,2,3,3,5,4,4,4,2,3,2,1,3,4,3,3,2,2,3,4,4,4,2,2,2,2,5,5,

%U 5,4,4,4,2,4,5,3,3,2,2,5

%N Number of ways to write n as p + x^3 + y^4, where p is a prime, and x and y are nonnegative integers.

%C Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 9, 10, 16, 17, 36, 52, 502.

%C (ii) Any integer n > 1 can be written as p + x^3 + 2*y^3, where p is a prime, and x and y are nonnegative integers.

%C (iii) Any integer n > 2 can be written as p + ((q-1)/2)^2 + x^4, where p and q are primes, and x is a nonnegative integer.

%C (iv) Any integer n > 5 can be written as p + q^2 + x^2, where p and q are primes, and x is a nonnegative integer.

%C (v) Any integer n > 5 can be written as p + q^2 + ((r-3)/2)^3, where p and q are primes, and r is an odd prime.

%C Ju. V. Linnik proved in 1960 that any sufficiently large integer can be written as the sum of a prime and two squares.

%H Zhi-Wei Sun, <a href="/A276415/b276415.txt">Table of n, a(n) for n = 1..10000</a>

%H Ju. V. Linnik, <a href="http://mi.mathnet.ru/eng/izv3674">An asymptotic formula in an additive problem of Hardy-Littlewood</a>, Izv. Akad. Nauk SSSR Ser. Mat., 24 (1960), 629-706 (Russian).

%e a(2) = 1 since 2 = 2 + 0^3 + 0^4 with 2 prime.

%e a(6) = 2 since 6 = 5 + 0^3 + 1^4 = 5 + 1^3 + 0^4 with 5 prime.

%e a(9) = 1 since 9 = 7 + 1^3 + 1^4 with 7 prime.

%e a(10) = 1 since 10 = 2 + 2^3 + 0^4 with 2 prime.

%e a(16) = 1 since 16 = 7 + 2^3 + 1^4 with 7 prime.

%e a(17) = 1 since 17 = 17 + 0^3 + 0^4 with 17 prime.

%e a(36) = 1 since 36 = 19 + 1^3 + 2^4 with 19 prime.

%e a(52) = 1 since 52 = 43 + 2^3 + 1^4 with 43 prime.

%e a(502) = 1 since 502 = 421 + 0^3 + 3^4 with 421 prime.

%p N:= 1000: # to get a(1) to a(N)

%p A:= Vector(N):

%p for p in select(isprime, [2,seq(i,i=3..N,2)]) do

%p   for x from 0 while p + x^3 <= N do

%p     for y from 0 while p + x^3 + y^4 <= N do

%p        r:= p+x^3+y^4;

%p        A[r]:= A[r]+1

%p od od od:

%p convert(A,list); # _Robert Israel_, Oct 05 2016

%t Do[r=0;Do[If[PrimeQ[n-x^3-y^4],r=r+1],{x,0,n^(1/3)},{y,0,(n-x^3)^(1/4)}];Print[n," ",r];Continue,{n,1,80}]

%Y Cf. A000040, A000290, A000578, A000583.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Sep 27 2016