[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,
made have errors):
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
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