A276415 Number of ways to write n as p + x^3 + y^4, where p is a prime, and x and y are nonnegative integers.

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, 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, 5, 4, 4, 4, 2, 4, 5, 3, 3, 2, 2, 5

Offset: 1

Author

Zhi-Wei Sun, Sep 27 2016

Keywords

• nonn

Comments

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

(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.

(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.

(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.

(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.

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

Examples

			a(2) = 1 since 2 = 2 + 0^3 + 0^4 with 2 prime.
a(6) = 2 since 6 = 5 + 0^3 + 1^4 = 5 + 1^3 + 0^4 with 5 prime.
a(9) = 1 since 9 = 7 + 1^3 + 1^4 with 7 prime.
a(10) = 1 since 10 = 2 + 2^3 + 0^4 with 2 prime.
a(16) = 1 since 16 = 7 + 2^3 + 1^4 with 7 prime.
a(17) = 1 since 17 = 17 + 0^3 + 0^4 with 17 prime.
a(36) = 1 since 36 = 19 + 1^3 + 2^4 with 19 prime.
a(52) = 1 since 52 = 43 + 2^3 + 1^4 with 43 prime.
a(502) = 1 since 502 = 421 + 0^3 + 3^4 with 421 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Ju. V. Linnik, An asymptotic formula in an additive problem of Hardy-Littlewood, Izv. Akad. Nauk SSSR Ser. Mat., 24 (1960), 629-706 (Russian).

Crossrefs

Cf. A000040, A000290, A000578, A000583.

Programs

Maple

N:= 1000: # to get a(1) to a(N)
A:= Vector(N):
for p in select(isprime, [2,seq(i,i=3..N,2)]) do
  for x from 0 while p + x^3 <= N do
    for y from 0 while p + x^3 + y^4 <= N do
       r:= p+x^3+y^4;
       A[r]:= A[r]+1
od od od:
convert(A,list); # Robert Israel, Oct 05 2016

Mathematica

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}]