A007416 The minimal numbers: sequence A005179 arranged in increasing order. (Formerly M1022)

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

1,2

Comments

Numbers k such that there is no x < k such that A000005(x) = A000005(k). - Benoit Cloitre, Apr 28 2002

A047983(a(n)) = 0. - Reinhard Zumkeller, Nov 03 2015

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

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.

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

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 *)

Prog

(PARI) for(s=1, 10^6, if(abs(prod(i=1, s-1, numdiv(i)-numdiv(s)))>0, print1(s, ", ")))
(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
(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
(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

Crossrefs

Subsequence of A025487; A002182 is a subsequence.

Cf. A005179, A099317, A099312, A099314, A099316, A099318.

Cf. A000005, A047983, A166721 (subsequence of squares).

Cf. A053212 and A064787 (the sequence {A000005(a(n))} and its inverse permutation).

Sequence in context: A089696 A171609 A099316 * A293132 A098895 A266543

Adjacent sequences: A007413 A007414 A007415 * A007417 A007418 A007419

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved