|
|
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.
|
|
2
|
|
|
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, 1, 1, 1, 0, 1, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,8
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
FORMULA
|
|
|
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
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|