A332765 - OEIS

%I #104 Dec 21 2020 07:23:47

%S 6,10,15,22,35,55,77,91,143,187,221,253,323,391,493,551,667,713,899,

%T 1073,1189,1271,1517,1591,1763,1961,2183,2419,2537,2773,3127,3233,

%U 3599,3953,4189,4331,4757,4897,5293,5723,5963,6499,6887,7171,7663,8051,8633,8989,9797,9991,10403,10807

%N 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 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 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 It appears that a(n) = A332877(n - 1) for n > 5.

%e Here are the ways (up to reversal) to order the first four primes:

%e   2, 3, 5, 7: Products: 6, 15, 35;  Largest product: 35

%e   2, 3, 7, 5: Products: 6, 21, 35;  Largest product: 35

%e   2, 5, 3, 7: Products: 10, 15, 21; Largest product: 21

%e   2, 5, 7, 3: Products: 10, 35, 21; Largest product: 35

%e   2, 7, 3, 5: Products: 14, 21, 15; Largest product: 21

%e   2, 7, 5, 3: Products: 14, 35, 15; Largest product: 35

%e   3, 2, 5, 7: Products: 6, 10, 35;  Largest product: 35

%e   3, 2, 7, 5: Products: 6, 14, 35;  Largest product: 35

%e   3, 5, 2, 7: Products: 15, 10, 14; Largest product: 15

%e   3, 7, 2, 5: Products: 21, 14, 10; Largest product: 21

%e   5, 2, 3, 7: Products: 10, 6, 21;  Largest product: 21

%e   5, 3, 2, 7: Products: 15, 6, 14;  Largest product: 15

%e The minimum largest product is 15, so a(4) = 15.

%e From _Gus Wiseman_, Dec 06 2020: (Start)

%e The sequence of terms together with their prime indices begins:

%e       6: {1,2}     551: {8,10}    3127: {16,17}

%e      10: {1,3}     667: {9,10}    3233: {16,18}

%e      15: {2,3}     713: {9,11}    3599: {17,18}

%e      22: {1,5}     899: {10,11}   3953: {17,19}

%e      35: {3,4}    1073: {10,12}   4189: {17,20}

%e      55: {3,5}    1189: {10,13}   4331: {18,20}

%e      77: {4,5}    1271: {11,13}   4757: {19,20}

%e      91: {4,6}    1517: {12,13}   4897: {17,23}

%e     143: {5,6}    1591: {12,14}   5293: {19,22}

%e     187: {5,7}    1763: {13,14}   5723: {17,25}

%e     221: {6,7}    1961: {12,16}   5963: {19,24}

%e     253: {5,9}    2183: {12,17}   6499: {19,25}

%e     323: {7,8}    2419: {13,17}   6887: {20,25}

%e     391: {7,9}    2537: {14,17}   7171: {20,26}

%e     493: {7,10}   2773: {15,17}   7663: {22,25}

%e (End)

%t primes[n_]:=Reverse[Prime/@Range[n]]; 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}]];a/@Range[2,53]

%t (* _Ivan N. Ianakiev_, Apr 28 2020 *)

%Y Cf. A332877, A333747.

%Y A338904 and A338905 have this sequence as row maxima.

%Y A339115 is the not necessarily squarefree version.

%Y A001358 lists semiprimes.

%Y A005117 lists squarefree numbers.

%Y A006881 lists squarefree semiprimes.

%Y A025129 gives the sum of squarefree semiprimes of weight n.

%Y A056239 (weight) gives the sum of prime indices of n.

%Y A320656 counts factorizations into squarefree semiprimes.

%Y A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.

%Y A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.

%Y A338907/A338908 list squarefree semiprimes of odd/even weight.

%Y A339114 is the least (squarefree) semiprime of weight n.

%Y A339116 groups squarefree semiprimes by greater prime factor.

%Y Cf. A001221, A014342, A024697, A046388, A062198, A098350, A112798, A168472, A320655, A338901.

%K nonn

%O 2,1

%A _Bobby Jacobs_, Apr 23 2020

%E a(12)-a(13) from _Jinyuan Wang_, Apr 24 2020

%E More terms from _Ivan N. Ianakiev_, Apr 28 2020