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Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).
3

%I #10 Jun 17 2022 08:34:45

%S 1,2,3,4,4,6,8,5,8,9,12,16,6,10,12,16,18,24,32,7,12,15,16,20,24,27,32,

%T 36,48,64,8,14,18,20,24,30,32,36,40,48,54,64,72,96,128,9,16,21,24,25,

%U 28,36,40,45,48,60,64,72,80,81,96,108,128,144,192,256

%N Irregular table read by rows: the n-th row gives the possible values of the number of divisors of numbers with n prime divisors (counted with multiplicity).

%C First differs from A074139 at the 8th row.

%C The n-th row begins with n+1, which corresponds to powers of primes, and ends with 2^n, which corresponds to squarefree numbers.

%C The n-th row contains the distinct values of the n-th row of A238963.

%e Table begins:

%e 1;

%e 2;

%e 3, 4;

%e 4, 6, 8;

%e 5, 8, 9, 12, 16;

%e 6, 10, 12, 16, 18, 24, 32;

%e 7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64;

%e 8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128;

%e ...

%e Numbers k with Omega(k) = 2 are either of the form p^2 with p prime, or of the form p1*p2 with p1 and p2 being distinct primes. The corresponding numbers of divisors are 3 and 4, respectively. Therefore the second row is {3, 4}.

%t row[n_] := Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[row, 9, 0] // Flatten

%o (PARI) row(n) = { my (m=Map()); forpart(p=n, mapput(m,prod(k=1, #p, 1+p[k]),0)); Vec(m) } \\ _Rémy Sigrist_, Jun 17 2022

%Y Cf. A000005, A001222, A036035, A063008, A074139, A238963, A355027 (row lengths).

%K nonn,tabf

%O 0,2

%A _Amiram Eldar_, Jun 16 2022