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

Author

Paolo Xausa, Aug 16 2021

Keywords

• nonn
• tabf

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).

Examples

			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
              ...
		

Crossrefs

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

Cf. A078721, A078722, A078724, A078725.

Programs

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}]

Formula

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