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

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 3360, 4320, 5280, 6240, 6720, 8160, 9120, 10080, 11040, 13440, 13920, 14880, 15840, 17760, 18720, 19680, 20160, 20640, 21600, 22560, 24480, 25440, 27360, 28320, 29280, 32160, 33120, 34080

Offset: 1

Author

J. Lowell, Aug 19 2008

Keywords

• nonn

Comments

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

All terms > 1 are even. A subsequence of A033950. - N. J. A. Sloane, Aug 27 2008

Contains 480*p for all primes p > 5 (see A109802). - N. J. A. Sloane, Aug 27 2008

Examples

			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.
		

References

• Dmitriy Kunisky, German Manoim and N. J. A. Sloane, On strongly refactorable numbers, in preparation.

Links

• German Manoim and N. J. A. Sloane, Sep 09 2008, Table of n, a(n) for n = 1..240937 [a large file]

Crossrefs

Cf. A033950, A134865, A109802, A141551, A141756, A141758, A141900, A142593, A142594, A100549, A100762, A082725, A135130, A143718, A143719, A143720.

Programs

Maple

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
## A100549: if n = prod_p p^e_p, then pp = largest prime <= 1 + max e_p
with(numtheory):
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;
isA141586 := proc(n) local ff,f,g,p,i; global pp;
ff := op(2..-1,ifactors(n));
for f in ff do
p := f[1];
if (add(floor(log(1+g[2])/log(p)),g in ff) > f[2]) then
return false;
end if;
end do;
for i from 1 to pi(pp(n)) do
p := ithprime(i);
if (n mod p <> 0) then
if (add(floor(log(1+g[2])/log(p)),g in ff) > 0) then
return false;
end if;
end if;
end do;
return true;
end proc; # David Applegate and N. J. A. Sloane, Sep 15 2008

Mathematica

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 *)
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 *)

PARI

is_A141586(n)={ bittest(n,0) & return(n==1); fordiv(n,d,n % numdiv(d) & return);1 } \\ M. F. Hasler, Dec 05 2010

Sage

is_A141586 = lambda n: all(number_of_divisors(d).divides(n) for d in divisors(n)) # D. S. McNeil, Dec 05 2010

Extensions

More terms from German Manoim (gerrymanoim(AT)gmail.com), Aug 27 2008