A113832
Triangle read by rows: row n (n>=2) gives a set of n primes with the property that the pairwise averages are all primes, having the smallest largest element.
Original entry on oeis.org
3, 7, 3, 7, 19, 3, 11, 23, 71, 5, 29, 53, 89, 113, 3, 11, 83, 131, 251, 383, 5, 29, 113, 269, 353, 449, 509, 5, 17, 41, 101, 257, 521, 761, 881, 23, 431, 503, 683, 863, 1091, 1523, 1871, 2963, 31, 1123, 1471, 1723, 3463, 3571, 4651, 5563, 5743, 6991
Offset: 2
Triangle begins:
3, 7
3, 7, 19
3, 11, 23, 71
5, 29, 53, 89, 113
3, 11, 83, 131, 251, 383
5, 29, 113, 269, 353, 449, 509
The set of primes generated by {5, 29, 53, 89, 113} is {17, 29, 41, 47, 59, 59, 71, 71, 83, 101}.
- Antal Balog, The prime k-tuplets conjecture on average, in "Analytic Number Theory" (eds. B. C. Berndt et al.) Birkhäuser, Boston, 1990, pp. 165-204. [Background]
See
A115631 for the case when all pairwise averages are distinct primes.
A115765
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.
Original entry on oeis.org
3, 7, 5, 17, 29, 5, 509, 1013, 1109
Offset: 2
The set of primes generated by {5, 17, 29} is {5, 11, 17, 17, 17, 23, 29}.
Triangle begins:
3, 7
5, 17, 29
5, 509, 1013, 1109
-
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]
A307246
Smallest k for which a set of n primes <= k exists so that the averages of all nonempty subsets are all distinct primes.
Original entry on oeis.org
2, 7, 67, 1277, 2484733
Offset: 1
For any set of n elements, there are 2^n - 1 nonempty subsets.
For n=3, consider the set {7, 19, 67}.
The averages of the 2^3 - 1 = 7 nonempty subsets are:
avg({7}) = 7
avg({19}) = 19
avg({67}) = 67
avg({7, 19}) = 13
avg({7, 67}) = 37
avg({19, 67}) = 43
avg({7, 19, 67}) = 31
All these averages are different primes, and no such set exists with the largest element < 67. Hence, a(3) = 67.
Sets which minimize the largest elements are:
n = 1 {2}
n = 2 {3, 7}
n = 3 {7, 19, 67}
n = 4 {5, 17, 89, 1277}
n = 5 {209173, 322573, 536773, 1217893, 2484733}
For n > 1, largest element of row n of
A113833.
Showing 1-3 of 3 results.
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