A333747 -id:A333747 - cp's OEIS frontend

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

Bobby Jacobs, Apr 23 2020

nonn

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

			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)

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.

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 *)

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

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

More terms from Ivan N. Ianakiev, Apr 28 2020

A332877 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.

Original entry on oeis.org

6, 15, 21, 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, 11303

Offset: 2

Bobby Jacobs, Apr 11 2020

nonn

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.

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

			Here are the different ways to arrange the first 4 primes in a circle.
  2-3
  | |  Products: 6, 21, 35, 10. Largest product: 35.
  5-7
.
  2-3
  | |  Products: 6, 15, 35, 14. Largest product: 35.
  7-5
.
  2-5
  | |  Products: 10, 15, 21, 14. Largest product: 21.
  7-3
The minimum largest product is 21, so a(4)=21.

Giovanni Resta, Examples for a(2)-a(22)

Cf. A064796, A332765, A333747.

primes[n_]:=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[2,60] (* Ivan N. Ianakiev, Apr 20 2020 *)

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

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

a(12)-a(13) from Michel Marcus, Apr 14 2020
a(14) from Alois P. Heinz, Apr 15 2020
a(15)-a(22) from Giovanni Resta, Apr 19 2020
More terms from Ivan N. Ianakiev, Apr 20 2020

cp's OEIS Frontend

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.

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