A129758 - OEIS

%I #12 May 27 2024 06:52:02

%S 3,3,5,7,11,7,17,17,19,31,29,19,41,47,47,43,61,59,67,61,59,71,67,89,

%T 97,101,79,89,103,113,107,127,131,139,151,127,137,167,167,163,149,163,

%U 167,157,199,163,197,181,227,227,211,239,251,257,257,229,271,269

%N 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.

%C 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

%F A078497(n)-prime(n)=prime(n)-a(n)=d. - _R. J. Mathar_, May 19 2007

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

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

%p 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

%t 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}]

%Y Cf. A078497, A071681, A078611.

%K easy,nonn

%O 1,1

%A _Giovanni Teofilatto_, May 15 2007

%E Edited and extended by _R. J. Mathar_, _Giovanni Teofilatto_ and _Stefan Steinerberger_, May 19 2007