A064821 -id:A064821 - cp's OEIS frontend

A335047 Maximum sum of primes (see Comments).

0, 3, 8, 17, 24, 37, 52, 69, 86, 107, 128, 153, 178, 207, 236, 269, 302, 339, 376, 417, 458, 503, 548, 597, 646, 699, 752, 809, 866, 927, 988, 1053, 1118, 1187, 1256, 1329, 1402, 1479, 1556, 1637, 1718, 1803, 1888, 1977, 2066, 2159, 2252, 2349, 2446, 2547, 2648

Offset: 1

Ivan N. Ianakiev, Jun 05 2020

nonn

Out of all permutations of the numbers 1..n such that the sum of all adjacent numbers is a prime (A064821) find the one with the maximum sum of the primes. a(n) is the respective maximum sum or equals zero if a permutation does not exist.

The sum of primes arising from a permutation of 1..n is always equal to n*(n+1) minus the values of the two endpoints, so for n > 6 it is probable that a(n) = n*(n+1) - (1+3) if n is odd and a(n) = n*(n+1) - (1+2) if n is even. - Giovanni Resta, Jun 05 2020

			For n = 4 there are 4 permutations: 1234, 1432, 3214, 3412. The one with the maximum sum of 17 (5+7+5) is 1432.

Cf. A064821, A335048.

p[n_]:=Permutations[Range[n]];f[n_]:=Max[Total/@Select[Table[Table[
p[n][[j,i]]+p[n][[j,i+1]],{i,1,Length[p[n][[j]]]-1}],{j,1,Length[p[n]]}],
AllTrue[#,PrimeQ]&]];f/@Range[7] (* slow, just for demo *)
G[n_] := G[n] = Reap[Do[If[PrimeQ[i + j], Sow[i <-> j]], {i, n}, {j, i-1}]][[2, 1]]; a[n_] := Block[{p = 1 + Boole@ OddQ@ n, ep, s}, ep = SortBy[ Select[ Tuples[ Range[1, n, p], 2], #[[1]] > #[[2]] &], Total]; s = SelectFirst[ ep, FindHamiltonianPath[ G[n], #[[1]], #[[2]]] != {} &, {}]; If[s == {}, 0, n (n + 1) - Total[s]]]; Array[a, 51] (* Giovanni Resta, Jun 05 2020 *)

More terms from Giovanni Resta, Jun 05 2020

A335048 Minimum sum of primes (see Comments).

Original entry on oeis.org

0, 3, 8, 13, 22, 31, 44, 57, 74, 91, 112, 133, 158, 183, 212, 241, 274, 307, 344, 381, 422, 463, 508, 553, 602, 651, 704, 757, 814, 871, 932, 993, 1058, 1123, 1192, 1261, 1334, 1407, 1484, 1561, 1642, 1723, 1808, 1893, 1982, 2071, 2164, 2257, 2354, 2451, 2552

Offset: 1

Ivan N. Ianakiev, Jun 05 2020

nonn

Out of all permutations of the numbers 1..n such that the sum of all adjacent numbers is a prime (A064821) find the one with the minimum sum of the primes. a(n) is the respective minimum sum or equals zero if a permutation does not exist.

The sum of primes arising from a permutation of 1..n is always equal to n*(n+1) minus the values of the two endpoints, so for n > 1 it is probable that a(n) = n*(n+1) - (n+n-2) if n is odd and a(n) = n*(n+1) - (n+n-1) if n is even. - Giovanni Resta, Jun 05 2020

			For n = 4 there are 4 permutations: 1234, 1432, 3214, 3412. The one with the minimum sum of 13 (5+3+5) is 3214.

Cf. A064821, A335047.

p[n_]:=Permutations[Range[n]];g[n_]:=Min[Total/@Select[Table[Table[
p[n][[j,i]]+p[n][[j,i+1]],{i,1,Length[p[n][[j]]]-1}],{j,1,Length[p[n]]}],AllTrue[#,PrimeQ]&]];g/@Range[7] (* slow, just for demo *)
G[n_] := G[n] = Reap[Do[If[PrimeQ[i + j], Sow[i <-> j]], {i, n}, {j, i-1}]][[2, 1]]; a[n_] := Block[{p = 1 + Boole@OddQ@n, ep, s}, ep = Reverse@ SortBy[ Select[ Tuples[ Range[1, n, p], 2], #[[1]] > #[[2]] &], Total]; s = SelectFirst[ ep, FindHamiltonianPath[G[n], #[[1]], #[[2]]] != {} &, {}]; If[s == {}, 0, n (n + 1) - Total[s]]]; Array[a, 51] (* Giovanni Resta, Jun 05 2020 *)

More terms from Giovanni Resta, Jun 05 2020

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A335047 Maximum sum of primes (see Comments).

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