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

1,3

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.

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).

Example

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.

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}]

Crossrefs

Cf. A000040, A000290, A000578, A000583.

Sequence in context: A202691 A328143 A278402 * A309262 A064983 A124933

Adjacent sequences: A276412 A276413 A276414 * A276416 A276417 A276418

Keyword

nonn

Author

Zhi-Wei Sun, Sep 27 2016

Status

approved