cp's OEIS Frontend

A355026 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).

%I A355026 #16 Jul 04 2025 03:08:27
%S A355026 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 A355026 36,48,64,8,14,18,20,24,30,32,36,40,48,54,64,72,96,128,9,16,21,24,25,
%U A355026 28,36,40,45,48,60,64,72,80,81,96,108,128,144,192,256
%N A355026 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 A355026 First differs from A074139 at the 8th row.
%C A355026 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 A355026 The n-th row contains the distinct values of the n-th row of A238963.
%H A355026 Amiram Eldar, <a href="/A355026/b355026.txt">Table of n, a(n) for n = 0..18645</a> (rows 0..32, flattened)
%e A355026 Table begins:
%e A355026   1;
%e A355026   2;
%e A355026   3, 4;
%e A355026   4, 6, 8;
%e A355026   5, 8, 9, 12, 16;
%e A355026   6, 10, 12, 16, 18, 24, 32;
%e A355026   7, 12, 15, 16, 20, 24, 27, 32, 36, 48, 64;
%e A355026   8, 14, 18, 20, 24, 30, 32, 36, 40, 48, 54, 64, 72, 96, 128;
%e A355026   ...
%e A355026 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 A355026 row[n_] := Union[Times @@ (# + 1) & /@ IntegerPartitions[n]]; Array[row, 9, 0] // Flatten
%o A355026 (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 A355026 Cf. A000005, A001222, A036035, A063008, A074139, A238963, A355027 (row lengths).
%K A355026 nonn,tabf
%O A355026 0,2
%A A355026 _Amiram Eldar_, Jun 16 2022