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A318277 Triangle read by rows; T(n, k) is the number of divisors of A025487(n) having the same prime signature as A025487(k) where 1 <= k <= n.

%I A318277 #31 Dec 31 2018 06:10:11
%S A318277 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,
%T A318277 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,
%U A318277 1,1,1,1,1,1,0,0,0,1,1,3,1,3,0,2,0,0,1,0,0,0,1,1,1,1,0,1,0,1
%N A318277 Triangle read by rows; T(n, k) is the number of divisors of A025487(n) having the same prime signature as A025487(k) where 1 <= k <= n.
%C A318277 If A025487(k) doesn't divide A025487(n) then T(n, k) = 0.
%C A318277 Adapted from _Clark Kimberling_ at A074206: "With different offset: A074206(A025487(n)) = sum of all A074206(A025487(k)) such that A025487(k) divides A025487(n) and A025487(k) < A025487(n)."
%C A318277 By looking at the number of divisors of A025487(n) that have the same prime signature as A025487(n) can help in computing A074206, especially if A025487(n) has a lot of divisors.
%H A318277 David A. Corneth, <a href="/A318277/b318277.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows, flattened)
%F A318277 Row sums are A000005(A025487(n)).
%e A318277 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.
%e A318277 Triangle with rows n and columns k starts:
%e A318277 1,
%e A318277 1, 1,
%e A318277 1, 1, 1,
%e A318277 1, 2, 0, 1,
%e A318277 1, 1, 1, 0, 1,
%e A318277 1, 2, 1, 1, 0, 1,
%e A318277 1, 1, 1, 0, 1, 0, 1,
%e A318277 1, 2, 1, 1, 1, 1, 0, 1,
%e A318277 1, 3, 0, 3, 0, 0, 0, 0, 1,
%e A318277 1, 1, 1, 0, 1, 0, 1, 0, 0, 1,
%e A318277 1, 2, 2, 1, 0, 2, 0, 0, 0, 0, 1,
%e A318277 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1,
%e A318277 1, 3, 1, 3, 0, 2, 0, 0, 1, 0, 0, 0, 1,
%t A318277 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 *)
%o A318277 (PARI) ps(y) = factor(y)[,2];
%o A318277 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
%Y A318277 Cf. A025487, A074206.
%K A318277 nonn,tabl
%O A318277 1,8
%A A318277 _David A. Corneth_, Aug 24 2018