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

Author

Giovanni Teofilatto, May 15 2007

Keywords

• easy
• nonn

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

Examples

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

Crossrefs

Cf. A078497, A071681, A078611.

Programs

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}]

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

Extensions

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