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

Author

Michel Marcus, Apr 22 2017

Keywords

• nonn
• tabf

Comments

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

Examples

			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;
...
		

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)

Crossrefs

Cf. A001221, A146289, A346617.

Programs

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 *)

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#v, v=concat(v,[1]), v[o]++); res=concat(res,v); i++); res} \\ David A. Corneth, Apr 22 2017

Formula

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