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A276237
Table read by antidiagonals: row n consists of k>=1 such that all primes dividing k divide n.
1
1, 1, 2, 1, 3, 4, 1, 2, 9, 8, 1, 5, 4, 27, 16, 1, 2, 25, 8, 81, 32, 1, 7, 3, 125, 16, 243, 64, 1, 2, 49, 4, 625, 32, 729, 128, 1, 3, 4, 343, 6, 3125, 64, 2187, 256, 1, 2, 9, 8, 2401, 8, 15625, 128, 6561, 512, 1, 11, 4, 27, 16, 16807, 9, 78125, 256, 19683, 1024
OFFSET
2,3
COMMENTS
If n is a power of prime p, row p consists of all powers of p.
Column 2 is A020639 (for n>=2).
LINKS
Robert Israel, Table of n, a(n) for n = 2..10012 (first 141 antidiagonals, flattened)
EXAMPLE
Table starts:
1 2 4 8 16 32 64 128 256
1 3 9 27 81 243 729 2187 6561
1 2 4 8 16 32 64 128 256
1 5 25 125 625 3125 15625 78125 390625
1 2 3 4 6 8 9 12 16
1 7 49 343 2401 16807 117649 823543 5764801
1 2 4 8 16 32 64 128 256
1 3 9 27 81 243 729 2187 6561
1 2 4 5 8 10 16 20 25
MAPLE
getRow:= proc(S, nk) option remember; local Q, x, count;
if nops(S) = 1 then [seq(S[1]^i, i=0..nk-1)]
elif nops(S) = 2 then
Q:= sort([seq(seq(S[1]^i*S[2]^j, i=0..nk-1), j=0..nk-1)]);
Q[1..nk]
else
Q:= NULL;
count:= 0;
for x from 1 while count < nk do
if numtheory:-factorset(x) subset S then
count:= count+1;
Q:= Q, x
fi
od;
[Q]
fi
end proc:
N:= 20: # for the first N-2 antidiagonals
A:= Matrix(N-1, N-2):
for n from 2 to N-1 do
A[n, 1..N-n]:= Vector[row](getRow(numtheory:-factorset(n), N-n))
od:
seq(seq(A[s-m, m], m=1..s-2), s=3..N);
MATHEMATICA
getRow[S_, nk_] := getRow[S, nk] = Module[{Q, x, count}, If[Length[S] == 1, Table[S[[1]]^i, {i, 0, nk - 1}], If[Length[S] == 2, Q = Sort[Flatten[ Table[Table[S[[1]]^i S[[2]]^j, {i, 0, nk - 1}], {j, 0, nk - 1}]]]; Q[[1 ;; nk]], Q = Nothing; count = 0; For[x = 1, count < nk, x++, If[ FactorInteger[x][[All, 1]] ~Subset~ S, count++; AppendTo[Q, x]]]]]];
M = 20; (* for the first M-2 antidiagonals *)
A = Array[0, {M - 1, M - 2}];
For[n = 2, n <= M - 1, n++, A[[n, 1 ;; M - n]] = getRow[FactorInteger[n][[All, 1]], M - n]];
Table[A[[s - m, m]], {s, 3, M}, {m, 1, s - 2}] // Flatten (* Jean-François Alcover, Oct 06 2020, after Robert Israel *)
CROSSREFS
Cf.: A003586 (row 6), A003592 (row 10).
Cf.: A020639.
Sequence in context: A372619 A330669 A084579 * A059663 A338367 A186975
KEYWORD
nonn,tabl
AUTHOR
Robert Israel, Dec 12 2016
STATUS
approved