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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. 25
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 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] EXAMPLE 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. 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 *) PROG (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 CROSSREFS Cf. A033950, A134865, A109802, A141551, A141756, A141758, A141900, A142593, A142594, A100549, A100762, A082725, A135130, A143718, A143719, A143720. Sequence in context: A262983 A324508 A100786 * A141758 A137496 A195015 Adjacent sequences:  A141583 A141584 A141585 * A141587 A141588 A141589 KEYWORD nonn AUTHOR J. Lowell, Aug 19 2008 EXTENSIONS More terms from German Manoim (gerrymanoim(AT)gmail.com), Aug 27 2008 STATUS approved

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Last modified October 14 05:08 EDT 2019. Contains 327995 sequences. (Running on oeis4.)