A376147 - OEIS

%I #22 Sep 19 2024 02:16:07

%S 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 40,32,33,22,50,45,36,26,49,56,60,48,39,44,70,63,64,54,17,55,105,84,

%U 75,72,65,52,66,112,100,80,81,19,77,88,98,125,120,90,51,34,78

%N 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 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 Michael De Vlieger, <a href="/A376147/b376147.txt">Table of n, a(n) for n = 1..12011</a> (rows n = 1..180, flattened)

%H Michael De Vlieger, <a href="/A376147/a376147.png">Log log scatterplot of a(n)</a>, n = 1..22572.

%e Construct the irregular table T(n,k) as follows:

%e 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 Read the table T(n,k) by antidiagonals downwards to obtain the data:

%e   2, 3, 4, 5, 8, 7, 15, 14, 21, 11, 35, 13.. (A056240)

%e            6, 9,10, 16, 20, 25, 28, 42, 22..

%e                 12, 18, 24, 30, 40, 50, 56..

%e                         27, 32, 45, 60, 63..

%e                             36, 48, 64, 75..

%e                                 54, 72, 80..

%e                                     81, 90..

%e And so on…

%t kk = 30;

%t MapIndexed[Set[t[First[#2]], #1] &,

%t   Rest@ CoefficientList[

%t     Series[(1 + x + 2 x^2 + x^4)/(1 - 3 x^3), {x, 0, kk}], x] ];

%t Array[Set[r[#],

%t   Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, t[kk]];

%t s = Table[

%t   Select[Range[Prime@ PrimePi[k], t[k]], r[#] == k &], {k, 2, kk}];

%t Join[{1}, s[[1]],

%t   Table[i = 1; m = n;

%t     Reap[While[And[m > 1, Length@ s[[m]] >= i], Sow[s[[m, i]] ]; m--;

%t i++]][[-1, 1]], {n, 2, kk - 1}] ] // Flatten (* _Michael De Vlieger_, Sep 18 2024 *)

%Y Cf. A000027, A000607, A000792, A001414, A002098, A056240, A064364.

%K nonn

%O 1,2

%A _David James Sycamore_, Sep 12 2024