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A218656 Number of ways to write 2n+1 as x+y with 0<x<y and x^4+y^4 prime. 8
1, 2, 3, 2, 3, 3, 1, 5, 4, 4, 4, 5, 4, 7, 6, 5, 3, 10, 4, 9, 8, 4, 9, 6, 7, 11, 7, 5, 11, 9, 9, 9, 11, 4, 14, 14, 9, 8, 9, 7, 11, 8, 12, 12, 10, 9, 11, 17, 10, 12, 16, 7, 13, 14, 8, 15, 9, 11, 23, 16, 9, 17, 23, 8, 15, 15, 11, 21, 18, 12, 19, 14, 15, 19, 21, 17, 16, 23, 13, 21, 20, 17, 29 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: a(n)>0 for all n=1,2,3,...

If we replace x^4+y^4 in the definition of a(n) by x^2+y^2, then a(n) was conjectured to be always positive by Thomas Ordowski on Nov 03 2012.

We also have similar conjectures with x^4+y^4 replaced by x^8+y^8 or x^{16}+y^{16}.

Alternate definition: Number of primes of the form k^4+(2n+1-k)^4, 0 < k <= n. - M. F. Hasler, Nov 05 2012

REFERENCES

Thomas Ordowski, Personal e-mail message, Nov 03 2012.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

EXAMPLE

For n=7 we have a(7)=1, since x^4+(15-x)^4 with 0<x<8 is prime only when x=4.

MAPLE

A218656 := n-> add(`if`(isprime(i^4+(2*n+1-i)^4), 1, 0), i=1..n): # Alois P. Heinz, Jul 09 2016

MATHEMATICA

a[n_]:=a[n]=Sum[If[PrimeQ[x^4+(2n+1-x)^4]==True, 1, 0], {x, 1, n}]

Do[Print[n, " ", a[n]], {n, 1, 20000}]

PROG

(PARI) A218586(n)=sum(x=1, n+0*n=2*n+1, isprime(x^4+(n-x)^4))  \\ - M. F. Hasler, Nov 05 2012

CROSSREFS

Cf. A002645, A218585, A218654.

Sequence in context: A138960 A245553 A115397 * A200323 A075370 A030350

Adjacent sequences:  A218653 A218654 A218655 * A218657 A218658 A218659

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 04 2012

STATUS

approved

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Last modified November 16 04:50 EST 2018. Contains 317252 sequences. (Running on oeis4.)