A332877 - OEIS

%I #91 Jul 05 2020 13:18:19

%S 6,15,21,35,55,77,91,143,187,221,253,323,391,493,551,667,713,899,1073,

%T 1189,1271,1517,1591,1763,1961,2183,2419,2537,2773,3127,3233,3599,

%U 3953,4189,4331,4757,4897,5293,5723,5963,6499,6887,7171,7663,8051,8633,8989,9797,9991,10403,10807,11303

%N Arrange the first n primes in a circle in any order. a(n) is the minimum value of the largest product of two consecutive primes out of all possible orders.

%C It might appear that all terms are either the product of two consecutive primes or two primes with a prime in between (A333747). However, 253=11*23 is the first term that is not in that sequence.

%C The easiest optimal permutation of n primes is probably {p_1, p_n, p_2, p_n-1, …, p_ceiling(n/2)}. - _Ivan N. Ianakiev_, Apr 20 2020

%H Giovanni Resta, <a href="/A332877/a332877.txt">Examples for a(2)-a(22)</a>

%F Probably a(n) = A332765(n+1) for n > 4.

%e Here are the different ways to arrange the first 4 primes in a circle.

%e   2-3

%e   | |  Products: 6, 21, 35, 10. Largest product: 35.

%e   5-7

%e .

%e   2-3

%e   | |  Products: 6, 15, 35, 14. Largest product: 35.

%e   7-5

%e .

%e   2-5

%e   | |  Products: 10, 15, 21, 14. Largest product: 21.

%e   7-3

%e The minimum largest product is 21, so a(4)=21.

%t primes[n_]:=Prime/@Range[n];

%t partition[n_]:=Partition[primes[n],UpTo[Ceiling[n/2]]];

%t riffle[n_]:=Riffle[partition[n][[1]],Reverse[partition[n][[2]]]];

%t a[n_]:=Max[Table[riffle[n][[i]]*riffle[n][[i+1]],{i,1,n-1}]];

%t a/@Range[2,60] (* _Ivan N. Ianakiev_, Apr 20 2020 *)

%o (PARI) a(n) = {my(x = oo); for (k=1, (n-1)!, my(vp = Vec(numtoperm(n, k-1))); vp = apply(x->prime(x), vp); x = min(x, max(vp[1]*vp[n-1], vecmax(vector(n-1, j, vp[j]*vp[j+1]))));); x;} \\ _Michel Marcus_, Apr 14 2020

%Y Cf. A064796, A332765, A333747.

%K nonn

%O 2,1

%A _Bobby Jacobs_, Apr 11 2020

%E a(12)-a(13) from _Michel Marcus_, Apr 14 2020

%E a(14) from _Alois P. Heinz_, Apr 15 2020

%E a(15)-a(22) from _Giovanni Resta_, Apr 19 2020

%E More terms from _Ivan N. Ianakiev_, Apr 20 2020