A306307 Numbers that are divisible by the number of their nontrivial divisors.

4, 6, 8, 9, 10, 12, 14, 20, 22, 24, 25, 26, 28, 30, 32, 34, 38, 42, 44, 46, 48, 49, 52, 54, 58, 60, 62, 66, 68, 74, 76, 78, 80, 81, 82, 86, 90, 92, 94, 102, 106, 112, 114, 116, 118, 121, 122, 124, 134, 138, 140, 142, 146, 148, 150, 158, 160, 164, 166, 168, 169, 172, 174

Offset: 1

Author

Todor Szimeonov, Feb 05 2019

Keywords

• nonn

Comments

We may define the number of divisors of a number n in four ways:

(1) A070824(n) = number of nontrivial or real divisors: 1 < d < n;

(2) variant of A032741(n) = number of small divisors: 1 and real divisors;

(3) A032741(n) = number of big or proper divisors: real divisors and n;

(4) A000005(n) = number of all divisors of n: 1, n and real divisors.

The case (1), divisibility through the number of nontrivial divisors, defines this sequence.

Examples

			1 and the prime numbers do not have any nontrivial divisors; A070824(n) is 0 for n=1 or a prime, and so they are not terms.
The only nontrivial divisor of 4 is 2, so A070824(4) = 1; 4 is divisible by 1, so 4 is a term.
A070824(15) = 2, and 15 is not divisible by 2, so 15 is not a term.
		

References

• T. Szimeonov, A számok [The numbers], Budapest, 2019, VVMA, 124 p.

Crossrefs

Cf. A070824, A054010, A033950.

Programs

Mathematica

seqQ[n_] := (nd = DivisorSigma[0, n] - 2) > 0 && Divisible[n, nd]; Select[Range[200], seqQ] (* Amiram Eldar, Mar 11 2019 *)

PARI

f(n) = if (n==1, 0, numdiv(n)-2); \\ A070824
isok(n) = f(n) && !frac(n/f(n)); \\ Michel Marcus, Feb 17 2019

Extensions

More terms from Michel Marcus, Feb 17 2019