A129758 Smallest prime p such that there are primes q and r with the property that p, q and r form an arithmetic progression and their sum is the same as three times the (n+2)-nd prime number.

3, 3, 5, 7, 11, 7, 17, 17, 19, 31, 29, 19, 41, 47, 47, 43, 61, 59, 67, 61, 59, 71, 67, 89, 97, 101, 79, 89, 103, 113, 107, 127, 131, 139, 151, 127, 137, 167, 167, 163, 149, 163, 167, 157, 199, 163, 197, 181, 227, 227, 211, 239, 251, 257, 257, 229, 271, 269

Offset

1,1

Comments

The same selection rule as in A078497 applies: if there is more than one prime triple (p,q=p+d,r=q+d) with p+q+r=A001748(n), take p from the triple with minimum d. - R. J. Mathar, May 19 2007

Links

Table of n, a(n) for n=1..58.

Formula

A078497(n)-prime(n)=prime(n)-a(n)=d. - R. J. Mathar, May 19 2007

Conjecture: Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n+2)) = 1. - Alain Rocchelli, May 01 2024

Example

3 + 5 + 7 = 15, which is three times the (1+2)th prime number. Thus a(1) = 3.

Maple

A129758 := proc(n) local p3, i, d, r, p; p3 := ithprime(n) ; i := n+1 ; while true do r := ithprime(i) ; d := r-p3 ; p := p3-d ; if isprime(p) then RETURN(p) ; fi ; i := i+1 ; od ; RETURN(-1) ; end: for n from 3 to 60 do printf("%d, ", A129758(n)) ; od ; # R. J. Mathar, May 19 2007

Mathematica

a[n_]:=Module[{}, k=1; While[Not[PrimeQ[Prime[n+1]-k] && PrimeQ[Prime[n+1]+k]], k++ ]; Prime[n + 1] - k]; Table[a[n], {n, 2, 60}]

Crossrefs

Cf. A078497, A071681, A078611.

Sequence in context: A086341 A176513 A128424 * A176347 A161834 A141867

Adjacent sequences: A129755 A129756 A129757 * A129759 A129760 A129761

Keyword

easy,nonn

Author

Giovanni Teofilatto, May 15 2007

Extensions

Edited and extended by R. J. Mathar, Giovanni Teofilatto and Stefan Steinerberger, May 19 2007

Status

approved