A362641 - OEIS

%I #9 Apr 28 2023 19:45:19

%S 1,2,3,3,15,5,21,15,35,21,165,385,273,55,1001,39,2805,7735,133,561,

%T 13585,273,5865,124355,5187,1265,391391,741,27115,19605131,1767,64515,

%U 5766215,217,374187,12212915,313131,170085,142635185,63973,902451,13147103255,223041,101065,818183948197

%N Product of the smaller primes, p, in the Goldbach partitions of 2n such that p + q = 2n, p <= q, and p,q prime (or 1 if no Goldbach partition of 2n exists).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Product_{k=1..n} k^(c(k)*c(2n - k)), where c is the prime characteristic (A010051).

%F a(n) = Product_{p+q = 2n, p<=q, and p,q prime} p.

%F a(n) = A337568(n) / A362640(n).

%e a(10) = 21; 2*10 = 20 has two Goldbach partitions, namely 17+3 and 13+7. The product of the smaller parts of these partitions, is 3*7 = 21.

%t Table[Product[k^((PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1])), {k, n}], {n, 40}]

%Y Cf. A010051, A045917, A337568 (product of all prime parts), A362640 (product of the larger primes q).

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Apr 28 2023