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

%I A344482 #26 Aug 26 2021 10:44:57
%S A344482 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 A344482 53,31,59,37,61,41,67,43,71,47,73,79,53,83,59,89,61,97,67,101,71,103,
%U A344482 73,107,109,79,113,83,127,89,131,97,137,101,139,103,149,107,151
%N A344482 Primes, each occurring twice, such that a(C(n)) = a(4*n-C(n)) = prime(n), where C is the Connell sequence (A001614).
%C A344482 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 A344482 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 A344482 The triangle has the following properties.
%C A344482 Row lengths are the positive odd numbers (A005408).
%C A344482 First column is A078721.
%C A344482 Column 3 is A078722 (for n >= 1).
%C A344482 Column 5 is A078724 (for n >= 2).
%C A344482 Column 7 is A078725 (for n >= 3).
%C A344482 Each even column is equal to the column preceding it.
%C A344482 Row records (A011756) are in the right border.
%C A344482 Indices of row records are the positive terms of A000290.
%C A344482 Each row r contains r terms that are duplicated in the next row.
%C A344482 In each row, the sum of terms which are not already listed in the sequence give A007468.
%C A344482 For rows r >= 2, row sum is A007468(r)+A007468(r-1) and row product is A007467(r)*A007467(r-1).
%F A344482 a(A001614(n)) = a(4*n-A001614(n)) = prime(n).
%e A344482 Written as an irregular triangle the sequence begins:
%e A344482    2;
%e A344482    3,   2,   5;
%e A344482    7,   3,  11,   5,  13;
%e A344482   17,   7,  19,  11,  23,  13,  29;
%e A344482   31,  17,  37,  19,  41,  23,  43,  29,  47;
%e A344482   53,  31,  59,  37,  61,  41,  67,  43,  71,  47,  73;
%e A344482   79,  53,  83,  59,  89,  61,  97,  67, 101,  71, 103,  73, 107;
%e A344482   ...
%e A344482 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 A344482                 2
%e A344482              3  2  5
%e A344482           7  3 11  5 13
%e A344482       17  7 19 11 23 13 29
%e A344482    31 17 37 19 41 23 43 29 47
%e A344482               ...
%t A344482 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 A344482 (* Second program, triangle rows *)
%t A344482 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 A344482 Cf. A000040, A117384, A000290, A001614, A005408, A007467, A007468, A011756.
%Y A344482 Cf. A078721, A078722, A078724, A078725.
%K A344482 nonn,tabf
%O A344482 1,1
%A A344482 _Paolo Xausa_, Aug 16 2021