A306312 - OEIS

%I #13 Feb 06 2019 13:08:31

%S 1,2,2,3,2,3,2,3,3,3,2,4,2,3,3,3,2,4,2,4,3,3,2,4,3,3,3,4,2,4,2,3,3,3,

%T 3,5,2,3,3,4,2,4,2,4,4,3,2,4,3,4,3,4,2,4,3,4,3,3,2,5,2,3,4,3,3,4,2,4,

%U 3,4,2,5,2,3,4,4,3,4,2,4,3,3,2,5,3,3,3,4

%N Number of terms of the set of divisors of n that are not the product of any other two distinct divisors.

%C Sets contain 1, primes and powers of primes.

%C a(n) <= A000005(n), a(n) <= A222084(n) and a(p) = 2 with p prime.

%C Record values for:

%C a(1) = 1

%C a(2) = 2

%C a(4) = 3

%C a(12) = 4

%C a(36) = 5

%C a(180) = 6

%C a(900) = 7

%C a(6300) = 8

%C a(44100) = 9

%C a(485100) = 10, ...

%C Records are obtained for A061742 U A228593. - _David A. Corneth_, Feb 06 2019

%e Divisors of 198 are 1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198. Here the set is 1, 2, 3, 9, 11 because 2*3 = 6, 2*9 = 18, 2*11 = 22, 3*11 = 33, 6*11 = 66, 9*11 = 99, 2*99 = 198. Then a(198) = 5.

%p with(numtheory): with(combinat): P:=proc(q) local a,b,c,k,n;

%p for n from 2 to q do if isprime(n) then print(2) else a:=sort([op(divisors(n) minus {1})]); b:=choose(a,2); c:=[];

%p for k from 1 to nops(b) do c:=[op(c),b[k][1]*b[k][2]]; od;

%p a:=[1,op({op(a)} minus {op(c)})]; print(nops(a)); fi; od; end: P(10^6);

%o (PARI) a(n) = my(f = factor(n)[, 2]); sum(i = 1, #f, min(2, f[i])) \\ _David A. Corneth_, Feb 06 2019

%Y Cf. A000005, A061742, A222084, A228593.

%K nonn

%O 1,2

%A _Paolo P. Lava_, Feb 06 2019