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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..63.

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A070824, A054010, A033950.

Sequence in context: A128510 A109289 A066071 * A355170 A248010 A067013

Adjacent sequences: A306304 A306305 A306306 * A306308 A306309 A306310

Keyword

nonn

Author

Todor Szimeonov, Feb 05 2019

Extensions

More terms from Michel Marcus, Feb 17 2019

Status

approved