# [seqfan] Re: Sums of two primes with prime subscripts

Richard Mathar mathar at strw.leidenuniv.nl
Fri Mar 26 12:19:28 CET 2010

```http://list.seqfan.eu/pipermail/seqfan/2010-March/004185.html says

jvp> Half-sums (averages) of two primes with prime subscripts.
jvp>
jvp> {(A006450(i) + A006450(j))/2} = {(A000040(A000040(i)) + A000040(A000040(j)))/2}
jvp>
jvp> n	sum(n)	halfsum(n)  Note
jvp> 1	6	3	3 + 3
jvp> 2	8	4	3 + 5
jvp> 3	10	5	5 + 5
jvp> 4	14	7	3 + 11
jvp> 5	16	8	5 + 11
jvp> 6	20	10	3 + 17
jvp> 7	22	11	11 + 11 = 17 + 5
jvp> 8	28	14	11 + 17
jvp> 9	34	17	17 + 17 = 3 + 31
jvp> 10	36	18	5 + 31
jvp> 11	42	21	11 + 31
jvp> 12	44	22	3 + 41
jvp> 13	46	23	5 + 41

Sorted, the halfsums(n) would continue 24, 26, 29,31 ,32, 35,
36, 38, 39, 41, 42, 43, 44, 45, 47, 49, 50, 54, 56 ... here, not 48 , 26, 29,..:

3, 4, 5, 7, 8, 10, 11, 14, 17, 18, 21, 22, 23, 24, 26, 29, 31, 32, 35, 36, 38,
39, 41, 42, 43, 44, 45, 47, 49, 50, 54, 56, 57, 59, 60, 62, 63, 65, 66, 67,
69, 70, 71, 72, 75, 79, 80, 81, 83, 84, 87, 88, 91, 92, 93, 94, 95, 96, 97,
98, 99, 101, 104, 105, 107, 108, 109, 110, 111, 112

jvp> 14	14	48	17 + 31 = 7 + 41
jvp> 15	52	26	11 + 41
jvp> 16	58	29	17 + 41
<snip>

In Maple:

hfs := {} ;
for i from 1 to 100 do
for j from 1 to i do
ithprime(ithprime(i))+ithprime(ithprime(j)) ;
hfs := hfs union {%/2}
end do:
end do:
sort(hfs) ;

```