A007416 The minimal numbers: sequence A005179 arranged in increasing order.
1, 2, 4, 6, 12, 16, 24, 36, 48, 60, 64, 120, 144, 180, 192, 240, 360, 576, 720, 840, 900, 960, 1024, 1260, 1296, 1680, 2520, 2880, 3072, 3600, 4096, 5040, 5184, 6300, 6480, 6720, 7560, 9216, 10080, 12288, 14400, 15120, 15360, 20160, 25200, 25920, 27720, 32400, 36864, 44100
Offset: 1
References
- J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 86.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..100000 (first 1000 from T. D. Noe and to 10000 from David A. Corneth)
- Ron Brown, The minimal number with a given number of divisors, Journal of Number Theory 116:1 (2005), pp. 150-158.
- M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
- J. Roberts, Lure of the Integers, Annotated scanned copy of pp. 81, 86 with notes.
- Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.
Crossrefs
Programs
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Haskell
a007416 n = a007416_list !! (n-1) a007416_list = f 1 [] where f x ts = if tau `elem` ts then f (x + 1) ts else x : f (x + 1) (tau:ts) where tau = a000005' x -- Reinhard Zumkeller, Apr 18 2015 -
Maple
for n from 1 to 10^5 do t:= numtheory:-tau(n); if not assigned(B[t]) then B[t]:= n fi; od: sort(map(op,[entries(B)]));# Robert Israel, Nov 11 2015
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Mathematica
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 *)
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PARI
for(s=1,10^6,if(abs(prod(i=1,s-1,numdiv(i)-numdiv(s)))>0,print1(s,",")))
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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
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PARI
A283980(n,f=factor(n))=prod(i=1, #f~, my(p=f[i, 1]); if(p==2, 6, nextprime(p+1))^f[i, 2]) 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) 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) list(lim)=winnow(A025487do(logint(lim\1-1,2)+1),lim) \\ Charles R Greathouse IV, Nov 17 2022
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