A103274 Number of ways of writing prime(n) in the form 2*prime(i)+prime(j).

0, 0, 0, 1, 2, 2, 4, 2, 3, 4, 2, 4, 5, 4, 4, 5, 3, 3, 5, 4, 4, 5, 4, 7, 6, 6, 5, 6, 6, 8, 6, 6, 8, 5, 8, 6, 6, 9, 5, 9, 7, 6, 6, 7, 10, 7, 8, 8, 6, 9, 12, 10, 7, 7, 11, 8, 10, 8, 11, 12, 9, 10, 12, 8, 10, 14, 12, 12, 7, 9, 12, 12, 11, 13, 10, 10, 15, 12, 15, 11, 12, 9, 12, 12, 12, 14, 12, 14, 13

Offset

1,5

Comments

First nonzero entry is for n=4: prime(4)=7=prime(1)+2*prime(3)=2+3*3, hence a(4)=1. Also, a(5)=2 because 11=5+2*3=7+2*2 (two solutions). Note that a(n) is not monotonic. - Zak Seidov, Jan 21 2006

Marnell conjectures that a(n) > 0 for n > 3. I find no exceptions below 10^9. - Charles R Greathouse IV, May 04 2010

References

Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193-196.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Formula

a(n) = A046926(prime(n)). - David Wasserman, Oct 08 2005

Example

11=2*2+7=2*3+5, so a(5)=2
a(100)=13 because p(100)=541=p(i)+2*p(j) for 13 pairs {i, j}: {2, 57}, {17, 53}, {23, 50}, {41, 42}, {49, 37}, {52, 36}, {56, 34}, {69, 25}, {76, 22}, {81, 18}, {91, 12}, {92, 11}, {96, 8}; e.g. 541=prime(96)+2*prime(8)=503+2*19. - Zak Seidov, Jan 21 2006

Mathematica

Table[Function[q, Length@ Select[#, Function[s, And[Length@ s == 2, Length@ First@ s == 1, MemberQ[Last@ , 2], Length@ Last@ s == 2]]] &@ Map[SortBy[Flatten[FactorInteger[#] /. {{p_, e_} /; e > 1 :> ConstantArray[p, e], {p_, 1} /; p > 1 :> p, {1, 1} -> 1}] & /@ #, Length] &, Select[IntegerPartitions[q, {2}], And[! MemberQ[#, 1], Total@ Boole@ PrimeQ@ # == 1] &]]]@ Prime@ n, {n, 89}] (* Michael De Vlieger, May 01 2017 *)

Prog

(PARI) a(n, q=prime(n))=my(s); forprime(p=2, q\2-1, if(isprime(q-2*p), s++)); s \\ Charles R Greathouse IV, Jul 22 2015

Crossrefs

Sequence in context: A163371 A061338 A135714 * A046820 A356878 A043262

Adjacent sequences: A103271 A103272 A103273 * A103275 A103276 A103277

Keyword

nonn

Author

Yasutoshi Kohmoto, Jan 27 2005

Extensions

More terms from David Wasserman, Oct 08 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007

Status

approved