A174682 -id:A174682 - cp's OEIS frontend

A175320 Positive integers which cannot be represented as half-sums (averages) of two primes with indices that are primes with prime indices.

1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85

Offset: 1

Jason Kimberley, Apr 04 2010

nonn

Appears to be finite with the last term being a(1578727)=161352166. If a(1578728) exists, it is greater than 10^9.
The (apparently) last 19 terms are listed in A175321.
Computed at the suggestion of Jonathan Vos Post.

J. S. Kimberley, Table of n, a(n) for n=1..1578727

Cf. A174682.

The complement of A175319.

A174681 Half-sums (averages) of two primes with prime subscripts.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Mar 26 2010

11 is the smallest value generated in two ways, as 22 is the smallest sum of two primes with prime subscripts in two ways: 11 + 11 = 17 + 5. Corrected and extended by R. J. Mathar, who also supplied the Maple code. Jason Kimberley also computed the first 733 positive integers which cannot be represented as the half sum of two primes with prime subscripts (A174682) as found using the first 998 values of A006450. From computing the first 20 thousand terms of A006450 (the 20000th term is 3118459), he shows the next value in the sequence of complements must be greater than 2907940. The PIP-Goldbach Conjecture is: every sufficiently large even number can be represented as the sum of two primes with prime subscripts.

			a(1) = 6/2 = 3 = (3 + 3)/2 because 3 is the first prime with prime subscript, p(2).
a(2) = 8/2 = 4 = (3 + 5)/2 because 5 is the second prime with prime subscript, p(3).
a(3) = 10/2 = 5 = (5 + 5)/2.
a(4) = 14/2 = 7 = (3 + 11)/2 because 11 is the second prime with prime subscript, p(5).

Cf. A000040, A006450, A174682.

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) ;

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

A303403 Even numbers that are not the sum of two prime-indexed primes.

Original entry on oeis.org

2, 4, 12, 18, 24, 26, 30, 32, 38, 40, 50, 54, 56, 60, 66, 68, 74, 80, 92, 96, 102, 104, 106, 110, 116, 122, 128, 136, 146, 148, 152, 154, 156, 164, 170, 172, 178, 180, 200, 204, 206, 212, 226, 230, 234, 248, 256, 260, 264, 268, 276, 290, 292, 296, 298, 302

Offset: 1

Amiram Eldar, May 13 2018

nonn

Bayless et al. conjectured that every even number larger than 80612 is the sum of two prime-indexed primes. If the conjecture is true then this sequence is finite with 733 terms.

Similarly, it appears that 322704332 is the largest of the 1578727 even numbers that cannot be written as prime(prime(prime(i))) + prime(prime(prime(j))). - Giovanni Resta, May 31 2018

			20 is not in the sequence since 20 = 17 + 3 = prime(7) + prime(2).  2 and 7 are primes, so 3 and 17 are prime-indexed primes. - _Michael B. Porter_, May 21 2018

Jonathan Bayless, Dominic Klyve, and Tomás Oliveira e Silva, New Bounds and Computations on Prime-Indexed Primes, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A43, 2013.

Cf. A006450, A014092, A166081.

Equals 2*A174682. - Michel Marcus, May 18 2018

pipQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]; s1falsifiziertQ[s_]:= Module[{ip=IntegerPartitions[s, {2}], widerlegt=False}, Do[If[pipQ[ip[[i, 1]] ] ~And~ pipQ [ip[[i, 2]] ], widerlegt = True; Break[]], {i, 1, Length[ip]}]; widerlegt]; Select[Range[2500],EvenQ[#]&& s1falsifiziertQ[ # ]==False&] (* after Michael Taktikos at A014092 *)
(* or *) p = Prime@ Prime@ Range@ PrimePi@ PrimePi@ 302; Select[Range[2, 302, 2], IntegerPartitions[#, {2}, p] == {} &] (* Giovanni Resta, May 31 2018 *)

isok(n) = {if (n % 2, return (0)); forprime(p=2, n/2, if (isprime(primepi(p)) && isprime(n-p) && isprime(primepi(n-p)), return (0));); return (1);} \\ Michel Marcus, May 18 2018

cp's OEIS Frontend

A175320 Positive integers which cannot be represented as half-sums (averages) of two primes with indices that are primes with prime indices.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A174681 Half-sums (averages) of two primes with prime subscripts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A303403 Even numbers that are not the sum of two prime-indexed primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI