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

1,2

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

Links

Table of n, a(n) for n=1..88.

Example

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.

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);

Prog

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

Crossrefs

Cf. A000005, A061742, A222084, A228593.

Sequence in context: A162361 A171135 A073855 * A350065 A077982 A331921

Adjacent sequences: A306309 A306310 A306311 * A306313 A306314 A306315

Keyword

nonn

Author

Paolo P. Lava, Feb 06 2019

Status

approved