A046926 -id:A046926

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A103274

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

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

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

A355484

a(n) is the least positive number that can be represented in exactly n ways as 2*p+q where p and q are primes.

+10
1

1, 6, 9, 21, 17, 33, 45, 51, 75, 99, 111, 93, 105, 135, 153, 201, 165, 249, 231, 237, 321, 225, 273, 363, 411, 393, 285, 315, 471, 483, 435, 405, 465, 555, 681, 495, 783, 675, 873, 849, 963, 1729, 585, 525, 897, 795, 1041, 915, 735, 855, 1191, 825, 765, 1095, 975, 1005, 1035, 1125, 1311, 1407

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) is the least number k such that A046926(k) = n.

LINKS

Robert Israel, Table of n, a(n) for n = 0..2500

EXAMPLE

a(3) = 21 because 21 can be written as 2*p+q with p and q prime in exactly 3 ways, namely 21 = 2*2+17 = 2*5+11 = 2*7+7, and no smaller number works.

MAPLE

M:= 3000: # to use primes up to M

P:= select(isprime, [2, seq(i, i=3..M, 2)]): nP:= nops(P):

A:= Vector(M):

for i from 1 do

p:= P[i];

if 2*p >= M then break fi;

for j from 1 to nP do

q:= P[j];

v:= 2*p+q;

if v > M then break fi;

A[v]:= A[v]+1;

od od:

K:= max(A):

V:= Array(0..K+1):

for i from 1 to M do

if V[A[i]] = 0 then V[A[i]]:= i fi

od:

L:= min(select(t -> V[t] = 0, [$1..K+1])):

convert(V[0..L-1], list);

CROSSREFS

Cf. A046926, A284052.