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%I #26 Aug 26 2021 10:44:57
%S 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,
%T 53,31,59,37,61,41,67,43,71,47,73,79,53,83,59,89,61,97,67,101,71,103,
%U 73,107,109,79,113,83,127,89,131,97,137,101,139,103,149,107,151
%N Primes, each occurring twice, such that a(C(n)) = a(4*n-C(n)) = prime(n), where C is the Connell sequence (A001614).
%C 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).
%C 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).
%C The triangle has the following properties.
%C Row lengths are the positive odd numbers (A005408).
%C First column is A078721.
%C Column 3 is A078722 (for n >= 1).
%C Column 5 is A078724 (for n >= 2).
%C Column 7 is A078725 (for n >= 3).
%C Each even column is equal to the column preceding it.
%C Row records (A011756) are in the right border.
%C Indices of row records are the positive terms of A000290.
%C Each row r contains r terms that are duplicated in the next row.
%C In each row, the sum of terms which are not already listed in the sequence give A007468.
%C For rows r >= 2, row sum is A007468(r)+A007468(r-1) and row product is A007467(r)*A007467(r-1).
%F a(A001614(n)) = a(4*n-A001614(n)) = prime(n).
%e Written as an irregular triangle the sequence begins:
%e 2;
%e 3, 2, 5;
%e 7, 3, 11, 5, 13;
%e 17, 7, 19, 11, 23, 13, 29;
%e 31, 17, 37, 19, 41, 23, 43, 29, 47;
%e 53, 31, 59, 37, 61, 41, 67, 43, 71, 47, 73;
%e 79, 53, 83, 59, 89, 61, 97, 67, 101, 71, 103, 73, 107;
%e ...
%e 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.
%e 2
%e 3 2 5
%e 7 3 11 5 13
%e 17 7 19 11 23 13 29
%e 31 17 37 19 41 23 43 29 47
%e ...
%t 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
%t (* Second program, triangle rows *)
%t 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}]
%Y Cf. A000040, A117384, A000290, A001614, A005408, A007467, A007468, A011756.
%Y Cf. A078721, A078722, A078724, A078725.
%K nonn,tabf
%O 1,1
%A _Paolo Xausa_, Aug 16 2021
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