OFFSET
1,8
COMMENTS
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
EXAMPLE
A025487(9) = 30 and A025487(4) = 6 and have prime signatures (1, 1, 1) and (1, 1) respectively. There are three divisors of 30 with the prime signature (1, 1), being 6, 10 and 15. Therefore, T(9, 4) = 3.
Triangle with rows n and columns k starts:
1,
1, 1,
1, 1, 1,
1, 2, 0, 1,
1, 1, 1, 0, 1,
1, 2, 1, 1, 0, 1,
1, 1, 1, 0, 1, 0, 1,
1, 2, 1, 1, 1, 1, 0, 1,
1, 3, 0, 3, 0, 0, 0, 0, 1,
1, 1, 1, 0, 1, 0, 1, 0, 0, 1,
1, 2, 2, 1, 0, 2, 0, 0, 0, 0, 1,
1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1,
1, 3, 1, 3, 0, 2, 0, 0, 1, 0, 0, 0, 1,
MATHEMATICA
f[n_] := Block[{lim, ww}, Set[{lim, ww}, {Product[Prime@ i, {i, n}], NestList[Append[#, 1] &, {1}, n - 1]} ]; {{{0}}}~Join~Map[Block[{w = #, k = 1}, Sort@ Apply[Join, {{ConstantArray[1, Length@ w]}, If[Length@ # == 0, #, #[[1]]] }] &@ Reap[Do[If[# <= lim, Sow[w]; k = 1, If[k >= Length@ w, Break[], k++]] &@ Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, #]] &@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]], {i, Infinity}]][[-1]]] &, ww]]; With[{s = Sort@ Map[{Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, #], #} &, Join @@ f@ 4]}, Table[DivisorSum[s[[n, 1]], 1 &, If[Length@ # == 1, #, TakeWhile[#, # > 0 &]] &@ Sort[If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #] &@ #, Greater] == s[[k, -1]] &], {n, Length@ s}, {k, n}]] // Flatten (* Michael De Vlieger, Oct 10 2018 *)
PROG
(PARI) ps(y) = factor(y)[, 2];
tabl(nn) = {v = al(nn); for (n=1, nn, d = divisors(v[n]); for (k=1, n, f = ps(v[k]); nb = #select(x->(ps(x) == f), d); print1(nb, ", "); ); print; ); } \\ Michel Marcus, Oct 11 2018; where al(n) is defined in A025487
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
David A. Corneth, Aug 24 2018
STATUS
approved