login
Triangle read by rows: row n (n>=2) gives a set of n primes with the property that the averages of all subsets are all primes, having the smallest largest element.
0

%I #6 Feb 03 2023 04:55:39

%S 3,7,5,17,29,5,509,1013,1109

%N Triangle read by rows: row n (n>=2) gives a set of n primes with the property that the averages of all subsets are all primes, having the smallest largest element.

%C See A113833 for the case of all subset averages being distinct primes. The Mathematica program is for row 4.

%H Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/PrimePatterns.pdf">Prime number patterns</a>

%e The set of primes generated by {5, 17, 29} is {5, 11, 17, 17, 17, 23, 29}.

%e Triangle begins:

%e 3, 7

%e 5, 17, 29

%e 5, 509, 1013, 1109

%t Needs["DiscreteMath`Combinatorica`"]; nn=PrimePi[1277]; Do[s=Prime[{l, k, j, i}]; ss=Rest[Subsets[s]]; ave=(Plus@@@ss)/(Length/@ss); If[And@@(IntegerQ/@ave) && And@@PrimeQ[ave], Break[]], {l, 2, nn}, {k, 2, l-1}, {j, 2, k-1}, {i, 2, j-1}]; Reverse[s]

%K nonn,tabf,hard,more

%O 2,1

%A _T. D. Noe_, Jan 30 2006