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.

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, 40, 32, 33, 22, 50, 45, 36, 26, 49, 56, 60, 48, 39, 44, 70, 63, 64, 54, 17, 55, 105, 84, 75, 72, 65, 52, 66, 112, 100, 80, 81, 19, 77, 88, 98, 125, 120, 90, 51, 34, 78

Offset: 1

Author

David James Sycamore, Sep 12 2024

Keywords

• nonn

Comments

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.

Examples

			Construct the irregular table T(n,k) as follows:
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).
Read the table T(n,k) by antidiagonals downwards to obtain the data:
  2, 3, 4, 5, 8, 7, 15, 14, 21, 11, 35, 13.. (A056240)
           6, 9,10, 16, 20, 25, 28, 42, 22..
                12, 18, 24, 30, 40, 50, 56..
                        27, 32, 45, 60, 63..
                            36, 48, 64, 75..
                                54, 72, 80..
                                    81, 90..
And so on…
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..12011 (rows n = 1..180, flattened)
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..22572.

Crossrefs

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

Programs

Mathematica

kk = 30;
MapIndexed[Set[t[First[#2]], #1] &,
  Rest@ CoefficientList[
    Series[(1 + x + 2 x^2 + x^4)/(1 - 3 x^3), {x, 0, kk}], x] ];
Array[Set[r[#],
  Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, t[kk]];
s = Table[
  Select[Range[Prime@ PrimePi[k], t[k]], r[#] == k &], {k, 2, kk}];
Join[{1}, s[[1]],
  Table[i = 1; m = n;
    Reap[While[And[m > 1, Length@ s[[m]] >= i], Sow[s[[m, i]] ]; m--;
i++]][[-1, 1]], {n, 2, kk - 1}] ] // Flatten (* Michael De Vlieger, Sep 18 2024 *)