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

Author

Yasutoshi Kohmoto, Jan 27 2005

Keywords

• nonn

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

Examples

			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
		

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

Programs

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 *)

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

Formula

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