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T(n,k) is the number of numbers <= 2^n having exactly k prime factors (with repetitions), 0<=k<=n, triangle read by rows.
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%I #15 Sep 23 2021 03:36:08

%S 1,1,1,1,2,1,1,4,2,1,1,6,6,2,1,1,11,10,7,2,1,1,18,22,13,7,2,1,1,31,42,

%T 30,14,7,2,1,1,54,82,60,34,15,7,2,1,1,97,157,125,71,36,15,7,2,1,1,172,

%U 304,256,152,77,37,15,7,2,1,1,309,589,513,325,168,81,37,15,7,2,1,1,564,1124,1049,669,367,177,83,37,15,7,2,1

%N T(n,k) is the number of numbers <= 2^n having exactly k prime factors (with repetitions), 0<=k<=n, triangle read by rows.

%H <a href="/index/Pri#primepop">Index entries for sequences related to numbers of primes in various ranges</a>

%F T(n,0) = 1; T(n,1) = A007053(n,1) for n>0; T(n,n) = 1.

%F Sum_{k=0..n} T(n,k) = 2^n.

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 4, 2, 1;

%e 1, 6, 6, 2, 1;

%e 1, 11, 10, 7, 2, 1;

%e 1, 18, 22, 13, 7, 2, 1;

%e ...

%t row[n_] := Module[{i, v = Table[0, {n}]}, For[i = 2, i <= 2^n, i++, v[[PrimeOmega[i]]]++]; Prepend[v, 1]];

%t Table[row[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Sep 23 2021, after PARI code *)

%o (PARI) row(n) = {v = vector(n); for (i=2, 2^n, v[bigomega(i)]++;); concat(1, v);}

%o tabl(nn) = for (n=0, nn, print(row(n))); \\ _Michel Marcus_, Aug 11 2015

%Y Cf. A001222.

%K nonn,tabl

%O 0,5

%A _Reinhard Zumkeller_, Mar 27 2004