1,2

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

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

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

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.

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)

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]

Cf. A033950, A134865, A109802, A141551, A141756, A141758, A141900, A142593, A142594.

Cf. A100549, A100762, A082725, A135130, A143718, A143719, A143720.

Sequence in context: A144551 A110821 A100786 this_sequence A141758 A137496 A051781

Adjacent sequences: A141583 A141584 A141585 this_sequence A141587 A141588 A141589

nonn

J. Lowell (jhbubby(AT)mindspring.com), Aug 19 2008

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