OFFSET
1,2
COMMENTS
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.
EXAMPLE
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…
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Sep 12 2024
STATUS
approved