# [seqfan] Sums of two primes with prime subscripts

Jonathan Post jvospost3 at gmail.com
Wed Mar 24 17:32:10 CET 2010

```Which of these would be preferred for OEIS (tables below hand-made,
Sums of two primes with prime subscripts.

Examples:
a(1) = 6 = 3 + 3 = prime(2) + prime(2) = prime(prime(1)) + prime(prime(1))
a(2) = 8 = 3 + 5 = prime(2) + prime(3) = prime(prime(1)) + prime(prime(2))

OR

Half-sums (averages) of two primes with prime subscripts.

{(A006450(i) + A006450(j))/2} = {(A000040(A000040(i)) + A000040(A000040(j)))/2}

n	sum(n)	halfsum(n)  Note
1	6	3	3 + 3
2	8	4	3 + 5
3	10	5	5 + 5
4	14	7	3 + 11
5	16	8	5 + 11
6	20	10	3 + 17
7	22	11	11 + 11 = 17 + 5
8	28	14	11 + 17
9	34	17	17 + 17 = 3 + 31
10	36	18	5 + 31
11	42	21	11 + 31
12	44	22	3 + 41
13	46	23	5 + 41
14	14	48	17 + 31 = 7 + 41
15	52	26	11 + 41
16	58	29	17 + 41
17	62	31	3 + 59 = 31 + 31
18	64	32	5 + 59
19	70	35	11 + 59 = 3 + 67
20	72	36	5 + 67
21	76	38	17 + 59
22	78	39	11 + 67
23	82	41	41 + 41
24	84	42	17 + 67
25	86	43	3 + 83
26	88	44	5 + 83
27	90	45	31 + 59
28	94	47	11 + 83
…

It appears that 22 is the smallest sum in two ways: 11 + 11 = 17 + 5.

One could also give the complements, i.e.:
Integers which cannot be represented as the half sum of two primes
with prime subscripts:
{1, 2, 6, 9, 12, 13, 15, 16, 19, 20, 25, 29, 30, 33, 34, 37, 39, 40, 46, …}

We certainly have:

11 is the largest number which cannot be represented by them sum of n
distinct primes with prime subscripts, because it is true that 11 is
the largest number which cannot be represented by them sum of n
distinct primes.  Hence in these sequences we restrict ourselves to
sums of two PIPs.

Once someone extends these sequences, by software, what is the
empirical status of:

The PIP-Goldbach Conjecture: every sufficiently large even number can
be represented as the sum of two primes with prime subscripts.

One can make a heuristic on the asymptotics of how sparse the primes
with prime subscripts less than a given N are, versus the number of
possible pairs of primes with prime subscripts that may be summed with
sum less than N.

JVP\ACP\Sum2PIPs.doc

```