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A376147 a(1)=1, followed by array T(n,k), n>=1, k>=2 read by antidiagonals (downwards) wherein the first row is A056240, and the k-th column records in ascending order the numbers m such that A001414(m) = k.

%I A376147 #22 Sep 19 2024 02:16:07
%S A376147 1,2,3,4,5,8,6,7,9,15,10,14,16,12,21,20,18,11,25,24,35,28,30,27,13,42,
%T A376147 40,32,33,22,50,45,36,26,49,56,60,48,39,44,70,63,64,54,17,55,105,84,
%U A376147 75,72,65,52,66,112,100,80,81,19,77,88,98,125,120,90,51,34,78
%N A376147 a(1)=1, followed by array T(n,k), n>=1, k>=2 read by antidiagonals (downwards) wherein the first row is A056240, and the k-th column records in ascending order the numbers m such that A001414(m) = k.
%C A376147 The columns of T(n,k) are of finite length, corresponding to A000607(k), whereas the rows are of infinite length. This is a permutation of A064364 (which reads 1 plus the consecutive columns of T(n,k)), and hence of the positive integers A000027.
%H A376147 Michael De Vlieger, <a href="/A376147/b376147.txt">Table of n, a(n) for n = 1..12011</a> (rows n = 1..180, flattened)
%H A376147 Michael De Vlieger, <a href="/A376147/a376147.png">Log log scatterplot of a(n)</a>, n = 1..22572.
%e A376147 Construct the irregular table T(n,k) as follows:
%e A376147 The first row T(1,k) is A056240, smallest number whose sum of prime divisors (with multiplicity) is k (k>=2).  The second row T(2,k) is the second smallest number (if it exists) whose sum of prime divisors is k, and so on. The k-th column is then the ordered list of the A000607(k) numbers (k>=2) whose prime divisors sum to k, the final term of which is A000792(k), after which the k-th column contains no further terms. The sum of the terms in the k-th column (k>=2) is A002098(k).
%e A376147 Read the table T(n,k) by antidiagonals downwards to obtain the data:
%e A376147   2, 3, 4, 5, 8, 7, 15, 14, 21, 11, 35, 13.. (A056240)
%e A376147            6, 9,10, 16, 20, 25, 28, 42, 22..
%e A376147                 12, 18, 24, 30, 40, 50, 56..
%e A376147                         27, 32, 45, 60, 63..
%e A376147                             36, 48, 64, 75..
%e A376147                                 54, 72, 80..
%e A376147                                     81, 90..
%e A376147 And so on…
%t A376147 kk = 30;
%t A376147 MapIndexed[Set[t[First[#2]], #1] &,
%t A376147   Rest@ CoefficientList[
%t A376147     Series[(1 + x + 2 x^2 + x^4)/(1 - 3 x^3), {x, 0, kk}], x] ];
%t A376147 Array[Set[r[#],
%t A376147   Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, t[kk]];
%t A376147 s = Table[
%t A376147   Select[Range[Prime@ PrimePi[k], t[k]], r[#] == k &], {k, 2, kk}];
%t A376147 Join[{1}, s[[1]],
%t A376147   Table[i = 1; m = n;
%t A376147     Reap[While[And[m > 1, Length@ s[[m]] >= i], Sow[s[[m, i]] ]; m--;
%t A376147 i++]][[-1, 1]], {n, 2, kk - 1}] ] // Flatten (* _Michael De Vlieger_, Sep 18 2024 *)
%Y A376147 Cf. A000027, A000607, A000792, A001414, A002098, A056240, A064364.
%K A376147 nonn
%O A376147 1,2
%A A376147 _David James Sycamore_, Sep 12 2024