cp's OEIS Frontend

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.

%I A276415 #26 Oct 06 2016 02:44:23
%S A276415 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 A276415 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 A276415 5,4,4,4,2,4,5,3,3,2,2,5
%N 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.
%C A276415 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 A276415 (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 A276415 (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 A276415 (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 A276415 (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 A276415 Ju. V. Linnik proved in 1960 that any sufficiently large integer can be written as the sum of a prime and two squares.
%H A276415 Zhi-Wei Sun, <a href="/A276415/b276415.txt">Table of n, a(n) for n = 1..10000</a>
%H A276415 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 A276415 a(2) = 1 since 2 = 2 + 0^3 + 0^4 with 2 prime.
%e A276415 a(6) = 2 since 6 = 5 + 0^3 + 1^4 = 5 + 1^3 + 0^4 with 5 prime.
%e A276415 a(9) = 1 since 9 = 7 + 1^3 + 1^4 with 7 prime.
%e A276415 a(10) = 1 since 10 = 2 + 2^3 + 0^4 with 2 prime.
%e A276415 a(16) = 1 since 16 = 7 + 2^3 + 1^4 with 7 prime.
%e A276415 a(17) = 1 since 17 = 17 + 0^3 + 0^4 with 17 prime.
%e A276415 a(36) = 1 since 36 = 19 + 1^3 + 2^4 with 19 prime.
%e A276415 a(52) = 1 since 52 = 43 + 2^3 + 1^4 with 43 prime.
%e A276415 a(502) = 1 since 502 = 421 + 0^3 + 3^4 with 421 prime.
%p A276415 N:= 1000: # to get a(1) to a(N)
%p A276415 A:= Vector(N):
%p A276415 for p in select(isprime, [2,seq(i,i=3..N,2)]) do
%p A276415   for x from 0 while p + x^3 <= N do
%p A276415     for y from 0 while p + x^3 + y^4 <= N do
%p A276415        r:= p+x^3+y^4;
%p A276415        A[r]:= A[r]+1
%p A276415 od od od:
%p A276415 convert(A,list); # _Robert Israel_, Oct 05 2016
%t A276415 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 A276415 Cf. A000040, A000290, A000578, A000583.
%K A276415 nonn
%O A276415 1,3
%A A276415 _Zhi-Wei Sun_, Sep 27 2016