A285577 Irregular triangle T(n,m) read by rows (n >= 1, 0 <= m <= Max(A001221([1..n]))), giving the number of integers in [1,n] with m distinct prime factors.

1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 4, 1, 1, 5, 1, 1, 6, 1, 1, 7, 1, 1, 7, 2, 1, 8, 2, 1, 8, 3, 1, 9, 3, 1, 9, 4, 1, 9, 5, 1, 10, 5, 1, 11, 5, 1, 11, 6, 1, 12, 6, 1, 12, 7, 1, 12, 8, 1, 12, 9, 1, 13, 9, 1, 13, 10, 1, 14, 10, 1, 14, 11, 1, 15, 11, 1, 15, 12, 1, 16, 12

Offset

1,5

Comments

A346617 is a similar triangle, except that the first column (corresponding to m = 0) has been omitted.

Links

Michel Marcus and David A. Corneth, Table of n, a(n) for n = 1..20001 (first 5062 rows flattened, first 806 terms from Michel Marcus)

Formula

See A346617 for the asymptotic distribution of the rows. - N. J. A. Sloane, Aug 19 2021

Example

First few rows are:
1;
1, 1;
1, 2;
1, 3;
1, 4;
1, 4, 1;
1, 5, 1;
1, 6, 1;
1, 7, 1;
1, 7, 2;
1, 8, 2;
...

Maple

omega := proc(n) nops(numtheory[factorset](n)) end proc: # # A001221
A:=Array(0..20, 0);
ans:=[];
mx:=0;
for n from 1 to 20 do
k:=omega(n);
if k>mx then mx:=k; fi;
A[k]:=A[k]+1;
ans:=[op(ans), [seq(A[i], i=0..mx)]];
od:
ans; # N. J. A. Sloane, Aug 19 2021

Mathematica

With[{nn = 29}, Function[s, Array[Function[t, Count[t, #] & /@ Range[0, Max@ t]]@ Take[s, #] &, nn]]@ PrimeNu@ Range@ nn] // Flatten (* Michael De Vlieger, Apr 23 2017 *)

Prog

(PARI) tabf(nn) = {for (n=1, nn, vo = vector(n, k, omega(k)); for (k=0, vecmax(vo), print1(#select(x->x==k, vo), ", "); ); print(); ); }
(PARI) upto(n) = {my(res = [1], v=[1], i=2); while(#res<n, o = omega(i)+1; if(o>#v, v=concat(v, [1]), v[o]++); res=concat(res, v); i++); res} \\ David A. Corneth, Apr 22 2017

Crossrefs

Cf. A001221, A146289, A346617.

Sequence in context: A351613 A094741 A360653 * A362470 A324392 A340273

Adjacent sequences: A285574 A285575 A285576 * A285578 A285579 A285580

Keyword

nonn,tabf

Author

Michel Marcus, Apr 22 2017

Status

approved