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A332765 Consider all permutations p_i of the first n primes; a(n) is the minimum over p_i of the maximal product of two adjacent primes in the permutation.

%I A332765 #104 Dec 21 2020 07:23:47
%S A332765 6,10,15,22,35,55,77,91,143,187,221,253,323,391,493,551,667,713,899,
%T A332765 1073,1189,1271,1517,1591,1763,1961,2183,2419,2537,2773,3127,3233,
%U A332765 3599,3953,4189,4331,4757,4897,5293,5723,5963,6499,6887,7171,7663,8051,8633,8989,9797,9991,10403,10807
%N A332765 Consider all permutations p_i of the first n primes; a(n) is the minimum over p_i of the maximal product of two adjacent primes in the permutation.
%C A332765 The optimal permutation of n primes is {p_n, p_1, p_n-1, p_2, …, p_ceiling(n/2)}. - _Ivan N. Ianakiev_, Apr 28 2020
%C A332765 Also the greatest squarefree semiprime whose prime indices sum to n + 1. A squarefree semiprime (A006881) is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798. - _Gus Wiseman_, Dec 06 2020
%F A332765 It appears that a(n) = A332877(n - 1) for n > 5.
%e A332765 Here are the ways (up to reversal) to order the first four primes:
%e A332765   2, 3, 5, 7: Products: 6, 15, 35;  Largest product: 35
%e A332765   2, 3, 7, 5: Products: 6, 21, 35;  Largest product: 35
%e A332765   2, 5, 3, 7: Products: 10, 15, 21; Largest product: 21
%e A332765   2, 5, 7, 3: Products: 10, 35, 21; Largest product: 35
%e A332765   2, 7, 3, 5: Products: 14, 21, 15; Largest product: 21
%e A332765   2, 7, 5, 3: Products: 14, 35, 15; Largest product: 35
%e A332765   3, 2, 5, 7: Products: 6, 10, 35;  Largest product: 35
%e A332765   3, 2, 7, 5: Products: 6, 14, 35;  Largest product: 35
%e A332765   3, 5, 2, 7: Products: 15, 10, 14; Largest product: 15
%e A332765   3, 7, 2, 5: Products: 21, 14, 10; Largest product: 21
%e A332765   5, 2, 3, 7: Products: 10, 6, 21;  Largest product: 21
%e A332765   5, 3, 2, 7: Products: 15, 6, 14;  Largest product: 15
%e A332765 The minimum largest product is 15, so a(4) = 15.
%e A332765 From _Gus Wiseman_, Dec 06 2020: (Start)
%e A332765 The sequence of terms together with their prime indices begins:
%e A332765       6: {1,2}     551: {8,10}    3127: {16,17}
%e A332765      10: {1,3}     667: {9,10}    3233: {16,18}
%e A332765      15: {2,3}     713: {9,11}    3599: {17,18}
%e A332765      22: {1,5}     899: {10,11}   3953: {17,19}
%e A332765      35: {3,4}    1073: {10,12}   4189: {17,20}
%e A332765      55: {3,5}    1189: {10,13}   4331: {18,20}
%e A332765      77: {4,5}    1271: {11,13}   4757: {19,20}
%e A332765      91: {4,6}    1517: {12,13}   4897: {17,23}
%e A332765     143: {5,6}    1591: {12,14}   5293: {19,22}
%e A332765     187: {5,7}    1763: {13,14}   5723: {17,25}
%e A332765     221: {6,7}    1961: {12,16}   5963: {19,24}
%e A332765     253: {5,9}    2183: {12,17}   6499: {19,25}
%e A332765     323: {7,8}    2419: {13,17}   6887: {20,25}
%e A332765     391: {7,9}    2537: {14,17}   7171: {20,26}
%e A332765     493: {7,10}   2773: {15,17}   7663: {22,25}
%e A332765 (End)
%t A332765 primes[n_]:=Reverse[Prime/@Range[n]]; partition[n_]:=Partition[primes[n],UpTo[Ceiling[n/2]]];
%t A332765 riffle[n_]:=Riffle[partition[n][[1]],Reverse[partition[n][[2]]]];
%t A332765 a[n_]:=Max[Table[riffle[n][[i]]*riffle[n][[i+1]],{i,1,n-1}]];a/@Range[2,53]
%t A332765 (* _Ivan N. Ianakiev_, Apr 28 2020 *)
%Y A332765 Cf. A332877, A333747.
%Y A332765 A338904 and A338905 have this sequence as row maxima.
%Y A332765 A339115 is the not necessarily squarefree version.
%Y A332765 A001358 lists semiprimes.
%Y A332765 A005117 lists squarefree numbers.
%Y A332765 A006881 lists squarefree semiprimes.
%Y A332765 A025129 gives the sum of squarefree semiprimes of weight n.
%Y A332765 A056239 (weight) gives the sum of prime indices of n.
%Y A332765 A320656 counts factorizations into squarefree semiprimes.
%Y A332765 A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
%Y A332765 A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
%Y A332765 A338907/A338908 list squarefree semiprimes of odd/even weight.
%Y A332765 A339114 is the least (squarefree) semiprime of weight n.
%Y A332765 A339116 groups squarefree semiprimes by greater prime factor.
%Y A332765 Cf. A001221, A014342, A024697, A046388, A062198, A098350, A112798, A168472, A320655, A338901.
%K A332765 nonn
%O A332765 2,1
%A A332765 _Bobby Jacobs_, Apr 23 2020
%E A332765 a(12)-a(13) from _Jinyuan Wang_, Apr 24 2020
%E A332765 More terms from _Ivan N. Ianakiev_, Apr 28 2020