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A141586 Strongly refactorable numbers: numbers n such that if n is divisible by d, it is divisible by the number of divisors of d.

%I A141586 #31 Apr 05 2019 10:44:33
%S A141586 1,2,12,24,36,72,240,480,720,1440,3360,4320,5280,6240,6720,8160,9120,
%T A141586 10080,11040,13440,13920,14880,15840,17760,18720,19680,20160,20640,
%U A141586 21600,22560,24480,25440,27360,28320,29280,32160,33120,34080
%N A141586 Strongly refactorable numbers: numbers n such that if n is divisible by d, it is divisible by the number of divisors of d.
%C A141586 Let n = Product_{p} p ^ e_p be the prime factorization of n and let M = max{e_p + 1 }. Then n is in the sequence iff for all primes q in the range 2 <= q <= M we have e_q >= Sum_{r} floor( log_q (e_r + 1) ). - _N. J. A. Sloane_, Sep 01 2008
%C A141586 All terms > 1 are even. A subsequence of A033950. - _N. J. A. Sloane_, Aug 27 2008
%C A141586 Contains 480*p for all primes p > 5 (see A109802). - _N. J. A. Sloane_, Aug 27 2008
%D A141586 Dmitriy Kunisky, German Manoim and N. J. A. Sloane, On strongly refactorable numbers, in preparation.
%H A141586 German Manoim and N. J. A. Sloane, Sep 09 2008, <a href="/A141586/b141586.txt">Table of n, a(n) for n = 1..240937</a> [a large file]
%e A141586 72 qualifies because its divisors are 1,2,3,4,6,8,9,12,18,24,36,72, which have 1,2,2,3,4,4,3,6,6,8,9,12 divisors respectively and all of those numbers are divisors of 72.
%p A141586 isA141586 := proc(n) local dvs,d ; dvs := numtheory[divisors](n) ; for d in dvs do if not numtheory[tau](d) in dvs then RETURN(false) : fi; od: RETURN(true) ; end: for n from 1 to 100000 do if isA141586(n) then printf("%d,",n) ; fi; od: # _R. J. Mathar_, Aug 26 2008
%p A141586 ## A100549: if n = prod_p p^e_p, then pp = largest prime <= 1 + max e_p
%p A141586 with(numtheory):
%p A141586 pp := proc(n) local f,m; option remember; if (n = 1) then return 1; end if; m := 1: for f in op(2..-1,ifactors(n)) do if (f[2] > m) then m := f[2]: end if; end do; prevprime(m+2); end proc;
%p A141586 isA141586 := proc(n) local ff,f,g,p,i; global pp;
%p A141586 ff := op(2..-1,ifactors(n));
%p A141586 for f in ff do
%p A141586 p := f[1];
%p A141586 if (add(floor(log(1+g[2])/log(p)),g in ff) > f[2]) then
%p A141586 return false;
%p A141586 end if;
%p A141586 end do;
%p A141586 for i from 1 to pi(pp(n)) do
%p A141586 p := ithprime(i);
%p A141586 if (n mod p <> 0) then
%p A141586 if (add(floor(log(1+g[2])/log(p)),g in ff) > 0) then
%p A141586 return false;
%p A141586 end if;
%p A141586 end if;
%p A141586 end do;
%p A141586 return true;
%p A141586 end proc; # _David Applegate_ and _N. J. A. Sloane_, Sep 15 2008
%t A141586 l = {}; For[n = 1, n < 100000, n++, b = DivisorSigma[0, Divisors[n]]; If[Length[Select[b, Mod[n, # ] > 0 &]] == 0, AppendTo[l, n]]]; l (* _Stefan Steinerberger_, Aug 25 2008 *)
%t A141586 sfnQ[n_]:=AllTrue[DivisorSigma[0,Divisors[n]],Mod[n,#]==0&]; Select[ Range[ 35000],sfnQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Jan 27 2019 *)
%o A141586 (PARI) is_A141586(n)={ bittest(n,0) & return(n==1); fordiv(n,d,n % numdiv(d) & return);1 } \\ _M. F. Hasler_, Dec 05 2010
%o A141586 (Sage) is_A141586 = lambda n: all(number_of_divisors(d).divides(n) for d in divisors(n)) # _D. S. McNeil_, Dec 05 2010
%Y A141586 Cf. A033950, A134865, A109802, A141551, A141756, A141758, A141900, A142593, A142594, A100549, A100762, A082725, A135130, A143718, A143719, A143720.
%K A141586 nonn
%O A141586 1,2
%A A141586 _J. Lowell_, Aug 19 2008
%E A141586 More terms from German Manoim (gerrymanoim(AT)gmail.com), Aug 27 2008