A115316 Lexicographically earliest permutation of the natural numbers such that each prime number is followed by exactly two composite numbers.

1, 2, 4, 6, 3, 8, 9, 5, 10, 12, 7, 14, 15, 11, 16, 18, 13, 20, 21, 17, 22, 24, 19, 25, 26, 23, 27, 28, 29, 30, 32, 31, 33, 34, 37, 35, 36, 41, 38, 39, 43, 40, 42, 47, 44, 45, 53, 46, 48, 59, 49, 50, 61, 51, 52, 67, 54, 55, 71, 56, 57, 73, 58, 60, 79, 62, 63, 83, 64, 65, 89, 66

Offset: 1

Author

Reinhard Zumkeller, Jan 20 2006

Keywords

• nonn
• easy

Comments

Inverse: A115318.

Fixed points = {1,2,27,28,29,30,33,34}, also for A115317, A115318, A115319.

Links

• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A002808, A018252, A073846.

Cf. A115317, A115318, A115319.

Programs

Mathematica

terms = 72;
np = Ceiling[terms/3] + 1;
nc = Ceiling[(2/3) terms];
pp = Prime[Range[np]];
cc = Partition[Select[Range[FindRoot[n == nc + PrimePi[n] + 1, {n, nc, 2nc}][[1, 2]] // Floor], CompositeQ], 2];
Join[{1}, Riffle[pp, cc] // Flatten][[1 ;; terms]] (* Jean-François Alcover, Nov 15 2021 *)

Formula

a(a(n)) = A115317(n).

a(3*n-2) = A018252(2*n-1), a(3*n-1) = A000040(n), a(3*n) = A018252(2*n);

a(n+1+floor((n+1)/2)) = A002808(n).