A115317 -id:A115317 - cp's OEIS frontend

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

Reinhard Zumkeller, Jan 20 2006

Inverse: A115318.

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

Index entries for sequences that are permutations of the natural numbers

Cf. A002808, A018252, A073846.

Cf. A115317, A115318, A115319.

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 *)

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).

A115319 a(n) = A115318(A115318(n)).

Original entry on oeis.org

1, 2, 8, 5, 6, 3, 14, 4, 11, 7, 12, 9, 20, 10, 17, 13, 18, 15, 26, 16, 23, 19, 25, 21, 22, 24, 27, 28, 29, 30, 31, 32, 33, 34, 37, 35, 36, 40, 42, 43, 39, 41, 38, 46, 48, 49, 45, 51, 54, 55, 57, 58, 44, 61, 63, 64, 53, 67, 52, 69, 47, 72, 56, 75, 76, 78, 60, 62, 81, 82, 50, 85

Offset: 1

Reinhard Zumkeller, Jan 20 2006

nonn

Inverse integer permutation of A115317.

Index entries for sequences that are permutations of the natural numbers

Cf. A115316, A115317, A115318.

nmax = 72;
nn = nmax + 15; (* estimated length of A115316 - should be increased in case of message "NotFound" in the output *)
np = Ceiling[nn/3] + 1;
nc = Ceiling[(2/3) nn];
pp = Prime[Range[np]];
cc = Partition[Select[Range[FindRoot[n == nc + PrimePi[n] + 1, {n, nc, 2nc}][[1, 2]] // Floor], CompositeQ], 2];
A115316 = Join[{1}, Riffle[pp, cc] // Flatten][[1 ;; nn]];
A115318[n_] := FirstPosition[A115316, n][[1]];
a[n_] := A115318[A115318[n]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Nov 15 2021 *)

A115316(a(n)) = a(A115316(n)) = A115318(n).

cp's OEIS Frontend

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

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