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A276415
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Number of ways to write n as p + x^3 + y^4, where p is a prime, and x and y are nonnegative integers.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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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).
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A000040, A000290, A000578, A000583.
Sequence in context: A202691 A328143 A278402 * A309262 A064983 A124933
Adjacent sequences: A276412 A276413 A276414 * A276416 A276417 A276418
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun, Sep 27 2016
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STATUS
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approved
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