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A141586
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Strongly refactorable numbers: numbers n with the property that if n is divisible by d, then n is divisible by the number of divisors of d.
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18
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| Let n = Prod_{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 (njas(AT)research.att.com), Sep 01 2008
All terms > 1 are even. A subsequence of A033950. - N. J. A. Sloane (njas(AT)research.att.com), Aug 27 2008
Contains 480*p for all primes p > 5 (see A109802). - N. J. A. Sloane (njas(AT)research.att.com), Aug 27 2008
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REFERENCES
| Dmitriy Kunisky, German Manoim and N. J. A. Sloane, On strongly refactorable numbers, in preparation.
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LINKS
| German Manoim and N. J. A. Sloane, Sep 09 2008, Table of n, a(n) for n = 1..240937 [a large file]
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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.
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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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 26 2008]
(From David Applegate and N. J. A. Sloane (njas(AT)research.att.com), Sep 15 2008. Start:) # 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; (End)
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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 [From Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Aug 25 2008]
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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 5 2010]
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CROSSREFS
| Cf. A033950, A134865, A109802, A141551, A141756, A141758, A141900, A142593, A142594.
Cf. A100549, A100762, A082725, A135130, A143718, A143719, A143720.
Sequence in context: A174457 A110821 A100786 * A141758 A137496 A195015
Adjacent sequences: A141583 A141584 A141585 * A141587 A141588 A141589
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KEYWORD
| nonn
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AUTHOR
| J. Lowell (jhbubby(AT)mindspring.com), Aug 19 2008
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EXTENSIONS
| More terms from German Manoim (gerrymanoim(AT)gmail.com), Aug 27 2008
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