A214834 -id:A214834 - cp's OEIS frontend

A156642 Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.

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

Offset: 0

Vladimir Shevelev, Feb 12 2009

nonn

Conjecture. For n >= 1, a(n) > 0. This conjecture does not follow from the validity of the Goldbach binary conjecture because numbers of the form 4n+2, generally speaking, also have decompositions into sums of two primes of the form 4k+1.

			From _Lei Zhou_, Mar 19 2013: (Start)
n=1: 4n+2=6, 6=3+3; this is the only case that matches the definition, so a(1)=1;
n=3: 4n+2=14, 14=3+11=7+7; two instances found, so a(3)=2. (End)

Lei Zhou, Table of n, a(n) for n = 0..10000

Cf. A002145, A002375, A061358, A002372, A045917, A214834, A217696.

Table[m = 4*n + 2; p1 = m + 1; ct = 0; While[p1 = p1 - 4; p2 = m - p1; p1 >= p2, If[PrimeQ[p1] && PrimeQ[p2], ct++]]; ct, {n, 1, 100}] (* Lei Zhou, Mar 19 2013 *)

A217696 Let p = A002145(n) be the n-th prime of the form 4k+3, then a(n) is the smallest number such that p is the smallest prime of the form 4k+3 for which 4*a(n)+2-p is prime.

Original entry on oeis.org

1, 4, 10, 24, 76, 102, 196, 74, 104, 348, 314, 345, 86, 660, 443, 1494, 914, 1329, 2613, 1635, 1316, 1856, 1688, 2589, 2628, 6423, 3116, 2165, 6320, 4445, 7278, 4743, 16539, 17783, 6084, 3806, 6281, 8946, 15129, 6266, 10976, 19538, 16794, 31160, 32916, 57041

Offset: 1

Lei Zhou, Mar 19 2013

nonn

It is conjectured that a(n) is defined for all positive integers.

This is also the index of first occurrence of the n-th prime in the form of 4k+3 in A214834.

			n=1: the first prime in the form of 4k+3 is 3, 3+3=6=4*1+2, so a(1)=1;
n=2: the second prime in the form of 4k+3 is 7, 7+7=14=3+11=4*3+2, and 11 is also a prime in the form of 4k+3, so a(2)!=3. 7+11=18=4*4+2=3+15, and 15 is not a prime number. So a(2)=4.

Lei Zhou, Table of n, a(n) for n = 1..170

Cf. A214834, A016825, A000040, A002145, A155642.

goal = 46; plst = {}; pct = 0; clst = {}; n = -1; While[pct < goal,
n = n + 4; If[PrimeQ[n], AppendTo[plst, n]; AppendTo[clst, 0];
  pct++]]; n = 2; cct = 0; While[cct < goal, n = n + 4; p1 = n + 1;
While[p1 = p1 - 4; p2 = n - p1; ! ((PrimeQ[p1]) && (PrimeQ[p2]) && (Mod[p2, 4] == 3))]; If[MemberQ[plst, p2], If[id = Position[plst, p2][[1, 1]]; clst[[id]] == 0, clst[[id]] = (n - 2)/4; cct++]]]; clst

ok(n,p)=if(!isprime(n-p),return(0));forprime(q=2,p-1,if(q%4==3 && isprime(n-q),return(0)));1
a(n)=my(p,k); forprime(q=2,,if(q%4==3&&n--==0,p=q;break)); k=(p+1)/4; while(!ok(4*k+2,p),k++); k \\ Charles R Greathouse IV, Mar 19 2013

A217697 a(n) is the smallest positive integer such that 4n+2 can be partitioned into the sum of two primes in the form of 4k+3 in n ways.

Original entry on oeis.org

1, 3, 12, 19, 28, 22, 37, 61, 58, 52, 67, 82, 124, 112, 148, 97, 175, 127, 214, 172, 157, 295, 280, 232, 217, 328, 331, 277, 247, 262, 520, 337, 388, 448, 430, 409, 382, 442, 367, 397, 610, 487, 412, 535, 547, 502, 592, 472, 703, 766, 652, 727, 637, 991, 802

Offset: 1

Lei Zhou, Mar 19 2013

nonn

This is also the index of the first occurrence of n in A156642.

			a(3) = 12 because 50 = 4*12 + 2 is the smallest number of the form 4m + 2 which can be expressed as a sum of 2 primes of the form 4k + 3 in 3 ways (3 + 47, 7 + 43, and 19 + 31).
a(3) = 12 because A156642(12) = 3 while for 0<=n<12, A156642(n) < 3.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A156642, A214834, A016825, A000040, A002145, A217696.

goal = 56; a = {}; Do[AppendTo[a, 0], {n, 1, goal}]; found = 0; k = 0; While[found < goal, k++; m = 4*k + 2; p1 = m + 1; ct = 0;
While[p1 = p1 - 4; p2 = m - p1; p1 >= p2, If[PrimeQ[p1] && PrimeQ[p2], ct++]]; If[ct <= goal, If[a[[ct]] == 0, a[[ct]] = k; found++]]]; a

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A156642 Number of decompositions of 4n+2 into unordered sums of two primes of the form 4k+3.

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A217696 Let p = A002145(n) be the n-th prime of the form 4k+3, then a(n) is the smallest number such that p is the smallest prime of the form 4k+3 for which 4*a(n)+2-p is prime.

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A217697 a(n) is the smallest positive integer such that 4n+2 can be partitioned into the sum of two primes in the form of 4k+3 in n ways.

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