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A364832
Rectangular array, read by descending antidiagonals, in which row n lists numbers m such that w(s,m) = n-1 (see comments for definition of w).
1
1, 3, 2, 7, 6, 4, 9, 14, 12, 5, 13, 18, 28, 15, 8, 19, 26, 36, 35, 24, 10, 21, 38, 52, 45, 56, 30, 11, 27, 42, 76, 65, 72, 70, 33, 16, 31, 54, 84, 95, 104, 90, 77, 48, 17, 37, 62, 108, 105, 152, 130, 99, 112, 51, 20, 39, 74, 124, 135, 168, 190, 143, 144, 119
OFFSET
1,2
COMMENTS
Let s be a set of primes. For each positive integer m, let w(s,m) = number of primes in s, not necessarily distinct, in the prime factorization of m. For the present sequence, we take s = A003627 = primes of the form 3k-1. Every positive integer occurs exactly once.
FORMULA
(row n) = A004612(n)*(row 1).
EXAMPLE
Corner:
1 3 7 9 13 19 21 27 31
2 6 14 18 26 38 42 54 62
4 12 28 36 52 76 84 108 124
5 15 35 45 65 95 105 135 155
8 24 56 72 104 152 168 216 248
10 30 70 90 130 190 210 270 310
11 33 77 99 143 209 231 297 341
16 48 112 144 208 304 336 432 496
MATHEMATICA
s = Select[Prime[Range[500]], Mod[#, 3] == 2 &]; (* A003627 *)
g[m_] := Select[FactorInteger[m], MemberQ[s, #[[1]]] &];
h := Times @@ (#[[1]]^#[[2]]) &[Transpose[#]] & ;
w[m_] := If[g[m] == {}, 1, h[g[m]]];
t = Table[w[m], {m, 1, 500}] ; (* A343430 *)
a[n_] := Flatten[Position[t, n]];
t1 = Select[Table[a[n], {n, 1, 30}], # != {} &];
TableForm[t1] (* this sequence as array *)
Table[t1[[n - k + 1, k]], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)
CROSSREFS
Cf. A000040, A003627, A004612 (column 1), A343430, A364808, A346414 (row 1).
Sequence in context: A268832 A201566 A072764 * A349890 A130328 A228993
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 09 2023
STATUS
approved