A344482 Primes, each occurring twice, such that a(C(n)) = a(4*n-C(n)) = prime(n), where C is the Connell sequence (A001614).

2, 3, 2, 5, 7, 3, 11, 5, 13, 17, 7, 19, 11, 23, 13, 29, 31, 17, 37, 19, 41, 23, 43, 29, 47, 53, 31, 59, 37, 61, 41, 67, 43, 71, 47, 73, 79, 53, 83, 59, 89, 61, 97, 67, 101, 71, 103, 73, 107, 109, 79, 113, 83, 127, 89, 131, 97, 137, 101, 139, 103, 149, 107, 151

Offset

1,1

Comments

Terms can be arranged in an irregular triangle read by rows in which row r is a permutation P of the primes in the interval [prime(s), prime(s+rlen-1)], where s = 1+(r-1)*(r-2)/2, rlen = 2*r-1 = A005408(r-1) and r >= 1 (see example).

P is the alternating (first term > second term < third term > fourth term < ...) permutation m -> 1, 1 -> 2, m+1 -> 3, 2 -> 4, m+2 -> 5, 3 -> 6, ..., rlen -> rlen where m = ceiling(rlen/2).

The triangle has the following properties.

Row lengths are the positive odd numbers (A005408).

First column is A078721.

Column 3 is A078722 (for n >= 1).

Column 5 is A078724 (for n >= 2).

Column 7 is A078725 (for n >= 3).

Each even column is equal to the column preceding it.

Row records (A011756) are in the right border.

Indices of row records are the positive terms of A000290.

Each row r contains r terms that are duplicated in the next row.

In each row, the sum of terms which are not already listed in the sequence give A007468.

For rows r >= 2, row sum is A007468(r)+A007468(r-1) and row product is A007467(r)*A007467(r-1).

Links

Table of n, a(n) for n=1..64.

Formula

a(A001614(n)) = a(4*n-A001614(n)) = prime(n).

Example

Written as an irregular triangle the sequence begins:
   2;
   3,   2,   5;
   7,   3,  11,   5,  13;
  17,   7,  19,  11,  23,  13,  29;
  31,  17,  37,  19,  41,  23,  43,  29,  47;
  53,  31,  59,  37,  61,  41,  67,  43,  71,  47,  73;
  79,  53,  83,  59,  89,  61,  97,  67, 101,  71, 103,  73, 107;
  ...
The triangle can be arranged as shown below so that, in every row, each odd position term is equal to the term immediately below it.
                2
             3  2  5
          7  3 11  5 13
      17  7 19 11 23 13 29
   31 17 37 19 41 23 43 29 47
              ...

Mathematica

nterms=64; a=ConstantArray[0, nterms]; For[n=1; p=1, n<=nterms, n++, If[a[[n]]==0, a[[n]]=Prime[p]; If[(d=4p-n)<=nterms, a[[d]]=a[[n]]]; p++]]; a
(* Second program, triangle rows *)
nrows=8; Table[rlen=2r-1; Permute[Prime[Range[s=1+(r-1)(r-2)/2, s+rlen-1]], Join[Range[2, rlen, 2], Range[1, rlen, 2]]], {r, nrows}]

Crossrefs

Cf. A000040, A117384, A000290, A001614, A005408, A007467, A007468, A011756.

Cf. A078721, A078722, A078724, A078725.

Sequence in context: A092556 A347000 A354271 * A092550 A058977 A337246

Adjacent sequences: A344479 A344480 A344481 * A344483 A344484 A344485

Keyword

nonn,tabf

Author

Paolo Xausa, Aug 16 2021

Status

approved