A355742 - OEIS

%I #8 Jul 21 2022 07:40:40

%S 1,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,3,0,2,0,2,0,1,0,1,0,2,0,1,0,1,0,

%T 2,0,3,0,2,0,1,0,2,0,1,0,2,0,4,0,1,0,4,0,1,0,3,0,1,0,3,0,2,0,2,0,1,0,

%U 2,0,3,0,2,0,1,0,2,0,2,0,1,0,1,0,1,0,2

%N Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cartesian_product">Cartesian product</a>.

%F Totally multiplicative with a(prime(k)) = A001222(k).

%e The prime indices of 49 are {4,4}, and the a(49) = 4 choices are: (2,2), (2,4), (4,2), (4,4).

%e The prime indices of 777 are {2,4,12}, and the a(777) = 6 choices are: (2,2,2), (2,2,3), (2,2,4), (2,4,2), (2,4,3), (2,4,4).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Times@@PrimeOmega/@primeMS[n],{n,100}]

%Y The unordered version is A001970, row-sums of A061260.

%Y Positions of 1's are A076610, just primes A355743.

%Y Positions of 0's are A299174.

%Y Allowing all divisors (not just primes) gives A355731, firsts A355732.

%Y Choosing only prime factors (not prime-powers) gives A355741.

%Y Counting multisets of primes gives A355744.

%Y The case of weakly increasing primes A355745, all divisors A355735.

%Y A000688 counts factorizations into prime powers.

%Y A001414 adds up distinct prime factors, counted by A001221.

%Y A003963 multiplies together the prime indices of n.

%Y A056239 adds up prime indices, row sums of A112798, counted by A001222.

%Y Cf. A000720, A120383, A279784, A289509, A324850, A355733, A355739, A355746.

%K nonn,mult

%O 1,7

%A _Gus Wiseman_, Jul 20 2022