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

%I A306307 #32 Mar 11 2019 08:22:13
%S A306307 4,6,8,9,10,12,14,20,22,24,25,26,28,30,32,34,38,42,44,46,48,49,52,54,
%T A306307 58,60,62,66,68,74,76,78,80,81,82,86,90,92,94,102,106,112,114,116,118,
%U A306307 121,122,124,134,138,140,142,146,148,150,158,160,164,166,168,169,172,174
%N A306307 Numbers that are divisible by the number of their nontrivial divisors.
%C A306307 We may define the number of divisors of a number n in four ways:
%C A306307 (1) A070824(n) = number of nontrivial or real divisors: 1 < d < n;
%C A306307 (2) variant of A032741(n) = number of small divisors: 1 and real divisors;
%C A306307 (3) A032741(n) = number of big or proper divisors: real divisors and n;
%C A306307 (4) A000005(n) = number of all divisors of n: 1, n and real divisors.
%C A306307 The case (1), divisibility through the number of nontrivial divisors, defines this sequence.
%D A306307 T. Szimeonov, A számok [The numbers], Budapest, 2019, VVMA, 124 p.
%e A306307 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.
%e A306307 The only nontrivial divisor of 4 is 2, so A070824(4) = 1; 4 is divisible by 1, so 4 is a term.
%e A306307 A070824(15) = 2, and 15 is not divisible by 2, so 15 is not a term.
%t A306307 seqQ[n_] := (nd = DivisorSigma[0, n] - 2) > 0 && Divisible[n, nd]; Select[Range[200], seqQ] (* _Amiram Eldar_, Mar 11 2019 *)
%o A306307 (PARI) f(n) = if (n==1, 0, numdiv(n)-2); \\ A070824
%o A306307 isok(n) = f(n) && !frac(n/f(n)); \\ _Michel Marcus_, Feb 17 2019
%Y A306307 Cf. A070824, A054010, A033950.
%K A306307 nonn
%O A306307 1,1
%A A306307 _Todor Szimeonov_, Feb 05 2019
%E A306307 More terms from _Michel Marcus_, Feb 17 2019