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.

6, 10, 15, 22, 35, 55, 77, 91, 143, 187, 221, 253, 323, 391, 493, 551, 667, 713, 899, 1073, 1189, 1271, 1517, 1591, 1763, 1961, 2183, 2419, 2537, 2773, 3127, 3233, 3599, 3953, 4189, 4331, 4757, 4897, 5293, 5723, 5963, 6499, 6887, 7171, 7663, 8051, 8633, 8989, 9797, 9991, 10403, 10807

Offset

2,1

Comments

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

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

Links

Table of n, a(n) for n=2..53.

Formula

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

Example

Here are the ways (up to reversal) to order the first four primes:
  2, 3, 5, 7: Products: 6, 15, 35;  Largest product: 35
  2, 3, 7, 5: Products: 6, 21, 35;  Largest product: 35
  2, 5, 3, 7: Products: 10, 15, 21; Largest product: 21
  2, 5, 7, 3: Products: 10, 35, 21; Largest product: 35
  2, 7, 3, 5: Products: 14, 21, 15; Largest product: 21
  2, 7, 5, 3: Products: 14, 35, 15; Largest product: 35
  3, 2, 5, 7: Products: 6, 10, 35;  Largest product: 35
  3, 2, 7, 5: Products: 6, 14, 35;  Largest product: 35
  3, 5, 2, 7: Products: 15, 10, 14; Largest product: 15
  3, 7, 2, 5: Products: 21, 14, 10; Largest product: 21
  5, 2, 3, 7: Products: 10, 6, 21;  Largest product: 21
  5, 3, 2, 7: Products: 15, 6, 14;  Largest product: 15
The minimum largest product is 15, so a(4) = 15.
From Gus Wiseman, Dec 06 2020: (Start)
The sequence of terms together with their prime indices begins:
      6: {1,2}     551: {8,10}    3127: {16,17}
     10: {1,3}     667: {9,10}    3233: {16,18}
     15: {2,3}     713: {9,11}    3599: {17,18}
     22: {1,5}     899: {10,11}   3953: {17,19}
     35: {3,4}    1073: {10,12}   4189: {17,20}
     55: {3,5}    1189: {10,13}   4331: {18,20}
     77: {4,5}    1271: {11,13}   4757: {19,20}
     91: {4,6}    1517: {12,13}   4897: {17,23}
    143: {5,6}    1591: {12,14}   5293: {19,22}
    187: {5,7}    1763: {13,14}   5723: {17,25}
    221: {6,7}    1961: {12,16}   5963: {19,24}
    253: {5,9}    2183: {12,17}   6499: {19,25}
    323: {7,8}    2419: {13,17}   6887: {20,25}
    391: {7,9}    2537: {14,17}   7171: {20,26}
    493: {7,10}   2773: {15,17}   7663: {22,25}
(End)

Mathematica

primes[n_]:=Reverse[Prime/@Range[n]]; partition[n_]:=Partition[primes[n], UpTo[Ceiling[n/2]]];
riffle[n_]:=Riffle[partition[n][[1]], Reverse[partition[n][[2]]]];
a[n_]:=Max[Table[riffle[n][[i]]*riffle[n][[i+1]], {i, 1, n-1}]]; a/@Range[2, 53]
(* Ivan N. Ianakiev, Apr 28 2020 *)

Crossrefs

Cf. A332877, A333747.

A338904 and A338905 have this sequence as row maxima.

A339115 is the not necessarily squarefree version.

A001358 lists semiprimes.

A005117 lists squarefree numbers.

A006881 lists squarefree semiprimes.

A025129 gives the sum of squarefree semiprimes of weight n.

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

A320656 counts factorizations into squarefree semiprimes.

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

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

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

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

A339116 groups squarefree semiprimes by greater prime factor.

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

Sequence in context: A049694 A285766 A157344 * A093773 A088708 A174872

Adjacent sequences: A332762 A332763 A332764 * A332766 A332767 A332768

Keyword

nonn

Author

Bobby Jacobs, Apr 23 2020

Extensions

a(12)-a(13) from Jinyuan Wang, Apr 24 2020

More terms from Ivan N. Ianakiev, Apr 28 2020

Status

approved