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A343809 Divide the positive integers into subsets of lengths given by successive primes, then reverse the order of terms in each subset.

%I A343809 #45 Sep 05 2023 12:22:08
%S A343809 2,1,5,4,3,10,9,8,7,6,17,16,15,14,13,12,11,28,27,26,25,24,23,22,21,20,
%T A343809 19,18,41,40,39,38,37,36,35,34,33,32,31,30,29,58,57,56,55,54,53,52,51,
%U A343809 50,49,48,47,46,45,44,43,42,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59
%N A343809 Divide the positive integers into subsets of lengths given by successive primes, then reverse the order of terms in each subset.
%C A343809 From _Omar E. Pol_, Apr 30 2021: (Start)
%C A343809 Irregular triangle read by rows T(n,k) in which row n lists the next p positive integers in decreasing order, where p is the n-th prime, with n >= 1.
%C A343809 The triangle has the following properties:
%C A343809 Column 1 gives the nonzero terms of A007504.
%C A343809 Column 2 gives A237589.
%C A343809 Column 3 gives A071148.
%C A343809 Column 4 gives the terms > 2 of A343859.
%C A343809 Column 5 gives the absolute values of the terms < -1 of A282329.
%C A343809 Column 6 gives the terms > 7 of A082548.
%C A343809 Column 7 gives the terms > 6 of A115030.
%C A343809 Records are in the column 1.
%C A343809 Indices of records are in the right border.
%C A343809 Right border gives A014284.
%C A343809 Row lengths give A000040.
%C A343809 Row products give A078423.
%C A343809 Row sums give A034956. (End)
%H A343809 Paolo Xausa, <a href="/A343809/b343809.txt">Table of n, a(n) for n = 1..10887</a> (rows n = 1..70 of triangle, flattened)
%H A343809 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A343809 T(n,k) = A007504(n) - k + 1, with n >= 1 and 1 <= k <= A000040(n). - _Omar E. Pol_, May 01 2021
%e A343809 From _Omar E. Pol_, Apr 30 2021: (Start)
%e A343809 Written as an irregular triangle in which row lengths give A000040 the sequence begins:
%e A343809    2,  1;
%e A343809    5,  4,  3;
%e A343809   10,  9,  8,  7,  6;
%e A343809   17, 16, 15, 14, 13, 12, 11;
%e A343809   28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18;
%e A343809   41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29;
%e A343809   58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42;
%e A343809   77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59;
%e A343809   ...
%e A343809 (End)
%p A343809 R:= NULL: t:= 1:
%p A343809 for i from 1 to 20 do
%p A343809   p:= ithprime(i);
%p A343809   R:= R, seq(i,i=t+p-1..t,-1);
%p A343809   t:= t+p;
%p A343809 od:
%p A343809 R; # _Robert Israel_, Apr 30 2021
%t A343809 With[{nn=10},Reverse/@TakeList[Range[Total[Prime[Range[nn]]]],Prime[Range[nn]]]]//Flatten (* _Harvey P. Dale_, Apr 27 2022 *)
%Y A343809 Cf. A000027, A000040, A007504, A014284, A034956, A038722, A071148, A073612 (fixed points), A078423, A082548, A115030, A237589, A282329, A343859, A344891.
%K A343809 nonn,tabf
%O A343809 1,1
%A A343809 _Paolo Xausa_, Apr 30 2021