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A007416 The minimal numbers: sequence A005179 arranged in increasing order.

%I A007416 M1022 #82 Nov 17 2022 10:02:00
%S A007416 1,2,4,6,12,16,24,36,48,60,64,120,144,180,192,240,360,576,720,840,900,
%T A007416 960,1024,1260,1296,1680,2520,2880,3072,3600,4096,5040,5184,6300,6480,
%U A007416 6720,7560,9216,10080,12288,14400,15120,15360,20160,25200,25920,27720,32400,36864,44100
%N A007416 The minimal numbers: sequence A005179 arranged in increasing order.
%C A007416 Numbers k such that there is no x < k such that A000005(x) = A000005(k). - _Benoit Cloitre_, Apr 28 2002
%C A007416 A047983(a(n)) = 0. - _Reinhard Zumkeller_, Nov 03 2015
%C A007416 Subsequence of A025487. If some m in A025487 is the first term in that sequence having its number of divisors, m is in this sequence. - _David A. Corneth_, Aug 31 2019
%D A007416 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 86.
%D A007416 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007416 Charles R Greathouse IV, <a href="/A007416/b007416.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 from T. D. Noe and to 10000 from David A. Corneth)
%H A007416 Ron Brown, <a href="https://doi.org/10.1016/j.jnt.2005.04.004">The minimal number with a given number of divisors</a>, Journal of Number Theory 116:1 (2005), pp. 150-158.
%H A007416 M. E. Grost, <a href="http://www.jstor.org/stable/2315183">The smallest number with a given number of divisors</a>, Amer. Math. Monthly, 75 (1968), 725-729.
%H A007416 J. Roberts, <a href="/A007415/a007415.pdf">Lure of the Integers</a>, Annotated scanned copy of pp. 81, 86 with notes.
%H A007416 Anna K. Savvopoulou and Christopher M. Wedrychowicz, <a href="http://dx.doi.org/10.1007/s11139-014-9572-9">On the smallest number with a given number of divisors</a>, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.
%p A007416 for n from 1 to 10^5 do
%p A007416   t:= numtheory:-tau(n);
%p A007416   if not assigned(B[t]) then B[t]:= n fi;
%p A007416 od:
%p A007416 sort(map(op,[entries(B)]));# _Robert Israel_, Nov 11 2015
%t A007416 A007416 = Reap[ For[ s = 1, s <= 10^5, s++, If[ Abs[ Product[ DivisorSigma[0, i] - DivisorSigma[0, s], {i, 1, s-1}]] > 0, Print[s]; Sow[s]]]][[2, 1]] (* _Jean-François Alcover_, Nov 19 2012, after Pari *)
%o A007416 (PARI) for(s=1,10^6,if(abs(prod(i=1,s-1,numdiv(i)-numdiv(s)))>0,print1(s,",")))
%o A007416 (PARI) is(n)=my(d=numdiv(n));for(i=1,n-1,if(numdiv(i)==d, return(0))); 1 \\ _Charles R Greathouse IV_, Feb 20 2013
%o A007416 (PARI)
%o A007416 A283980(n,f=factor(n))=prod(i=1, #f~, my(p=f[i, 1]); if(p==2, 6, nextprime(p+1))^f[i, 2])
%o A007416 A025487do(e) = my(v=List([1, 2]), i=2, u = 2^e, t); while(v[i] != u, if(2*v[i] <= u, listput(v, 2*v[i]); t = A283980(v[i]); if(t <= u, listput(v, t))); i++); Set(v)
%o A007416 winnow(v,lim=v[#v])=my(m=Map(),u=List()); for(i=1,#v, if(v[i]>lim, break); my(t=numdiv(v[i])); if(!mapisdefined(m,t), mapput(m,t,0); listput(u,v[i]))); m=0; Vec(u)
%o A007416 list(lim)=winnow(A025487do(logint(lim\1-1,2)+1),lim) \\ _Charles R Greathouse IV_, Nov 17 2022
%o A007416 (Haskell)
%o A007416 a007416 n = a007416_list !! (n-1)
%o A007416 a007416_list = f 1 [] where
%o A007416    f x ts = if tau `elem` ts then f (x + 1) ts else x : f (x + 1) (tau:ts)
%o A007416             where tau = a000005' x
%o A007416 -- _Reinhard Zumkeller_, Apr 18 2015
%Y A007416 Subsequence of A025487; A002182 is a subsequence.
%Y A007416 Cf. A005179, A099317, A099312, A099314, A099316, A099318.
%Y A007416 Cf. A000005, A047983, A166721 (subsequence of squares).
%Y A007416 Cf. A053212 and A064787 (the sequence {A000005(a(n))} and its inverse permutation).
%K A007416 nonn,easy,nice
%O A007416 1,2
%A A007416 _N. J. A. Sloane_