A243956 -id:A243956 - cp's OEIS frontend

A339625 a(n) is the number of ways to write 6*n = p + q with p a lesser twin prime (A001359) and q a greater twin prime (A006512).

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

Offset: 1

J. M. Bergot and Robert Israel, Dec 10 2020

nonn

If 6*n = p + q, then also 6*n = (p+2) + (q-2), with p+2 a greater and q-2 a lesser twin prime. Thus a(n) is odd if and only if n/2 is in A002822.

			a(4)=3 because 6*4 = 24 = 5 + 19 = 11 + 13 = 17 + 7 where (5,7), (11,13) and (17,19) are twin prime pairs.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A002822, A339630.

a(n)=0 for n in A243956.

N:= 600: # for a(1)..a(floor(N/6)))
P:= select(isprime, {seq(i,i=3..N,2)}):
T1:= sort(convert(P intersect map(`-`,P,2),list)):
T2:= map(`+`,T1,2):
V:= Vector(N):
nT:= nops(T1):
for i from 1 to nT do
  for j from 1 to nT do
    v:= T1[i]+T2[j];
    if v > N then break fi;
    V[v]:= V[v]+1;
od od:
seq(V[6*i],i=1..N/6);

A235644 Number of decompositions of 12*n into the sum of two (not necessarily distinct) twin prime pairs.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 0, 2, 1, 3, 3, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 3, 3, 6, 2, 3, 1, 2, 4, 3, 4, 4, 1, 3, 2, 3, 5, 2, 7, 1, 3, 2, 2, 5, 2, 5, 2, 3, 2, 2, 3, 5, 3, 4, 1, 0, 3, 1, 6, 2, 3, 3, 1, 5, 2, 5, 3, 3, 4, 1, 4

Offset: 1

Lear Young, Jun 16 2014

nonn

In the 1980's, Liang conjectured that (6n)^2 = p_1 + p_2 + p_3 + p_4, where (p_1, p_2) and (p_3, p_4) are twin prime pairs. See reference for more details.
It seems there are at least 2 solutions for the decompositions when n > 701.
If the two twin prime pairs are required to be distinct, the sequence is A187759.

			a(736) = 2 because 12*736 = 197 + 199 + 4217 + 4219 = 857 + 859 + 3557 + 3559, so there are 2 ways of expressing 12*n as the sum of two twin prime pairs.

Liang Ding Xiang, Problem 93#, Bulletin of Mathematics (Wuhan), 6(1992),41. ISSN 0488-7395.

Cf. A016910, A243941, A187759, A243914, A243956.

v=select(p->isprime(p-2)&&p>5, primes(200))\6; l=List(); for(i=1, #v, if(2*v[i]<100, listput(l, 2*v[i])); for(j=i+1, #v, if((v[i]+v[j])<100, listput(l, v[i]+v[j])))); l1=vecsort(l); k=1; for(i=1, 100, s=sum(j=k, #l1, l1[j]==i); print1(s", "); k+=s) \\ Lear Young, Jun 16 2014

v=select(p->isprime(p-2)&&p>5,primes(110))\6;for(i=1, 99, print1(sum(j=1,#v,vecsearch(v,i-v[j])>0 && i-v[j]>=v[j])", "))   \\ change i-v[j]>=v[j] to i-v[j]>v[j] is A187759.  Lear Young, Jun 16 2014

cp's OEIS Frontend

A339625 a(n) is the number of ways to write 6*n = p + q with p a lesser twin prime (A001359) and q a greater twin prime (A006512).

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