A306312 Number of terms of the set of divisors of n that are not the product of any other two distinct divisors.

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, 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, 3, 4, 2, 5, 2, 3, 4, 4, 3, 4, 2, 4, 3, 3, 2, 5, 3, 3, 3, 4

Offset: 1

Author

Paolo P. Lava, Feb 06 2019

Keywords

• nonn

Comments

Sets contain 1, primes and powers of primes.

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

Record values for:

a(1) = 1

a(2) = 2

a(4) = 3

a(12) = 4

a(36) = 5

a(180) = 6

a(900) = 7

a(6300) = 8

a(44100) = 9

a(485100) = 10, ...

Records are obtained for A061742 U A228593. - David A. Corneth, Feb 06 2019

Examples

			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.
		

Crossrefs

Cf. A000005, A061742, A222084, A228593.

Programs

Maple

with(numtheory): with(combinat): P:=proc(q) local a,b,c,k,n;
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:=[];
for k from 1 to nops(b) do c:=[op(c),b[k][1]*b[k][2]]; od;
a:=[1,op({op(a)} minus {op(c)})]; print(nops(a)); fi; od; end: P(10^6);

PARI

a(n) = my(f = factor(n)[, 2]); sum(i = 1, #f, min(2, f[i])) \\ David A. Corneth, Feb 06 2019