A099314 -id:A099314 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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.

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).

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

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

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 *)

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

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

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

A099316 Greatest 3-smooth number dividing the n-th minimal number.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 24, 36, 48, 12, 64, 24, 144, 36, 192, 48, 72, 576, 144, 24, 36, 192, 1024, 36, 1296, 48, 72, 576, 3072, 144, 4096, 144, 5184, 36, 1296, 192, 216, 9216, 288, 12288, 576, 432, 3072, 576, 144, 5184, 72, 1296, 36864, 36, 1296, 9216, 46656, 288

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416.

David A. Corneth, Table of n, a(n) for n = 1..20000 (first 1000 terms from Amiram Eldar)

Cf. A007416, A065331, A099315, A003586, A099312, A099314.

a(n) = A065331(A007416(n)).

A099312 Exponent of greatest power of 2 dividing the n-th minimal number.

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 2, 4, 2, 6, 3, 4, 2, 6, 4, 3, 6, 4, 3, 2, 6, 10, 2, 4, 4, 3, 6, 10, 4, 12, 4, 6, 2, 4, 6, 3, 10, 5, 12, 6, 4, 10, 6, 4, 6, 3, 4, 12, 2, 4, 10, 6, 5, 4, 6, 12, 16, 10, 3, 6, 10, 5, 6, 4, 4, 6, 12, 16, 6, 4, 10, 6, 18, 4, 10, 12, 5, 5, 10, 12, 4, 5, 16

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

A minimal number is the smallest number with a given number of divisors, see A007416.

David A. Corneth and Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007416, A007814, A099311, A099314, A099316.

a(n) = A007814(A007416(n)).

A099313 Exponent of greatest power of 3 dividing the smallest number having exactly n divisors.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 2, 4, 1, 2, 1, 0, 2, 0, 1, 2, 1, 4, 2, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 2, 6, 4, 2, 1, 0, 2, 4, 1, 2, 1, 0, 2, 0, 1, 2, 3, 4, 2, 0, 1, 2, 4, 0, 2, 0, 1, 4, 1, 6, 2, 0, 3, 2, 1, 0, 2, 4, 1, 2, 1, 0, 2, 6, 1, 2, 1, 4, 2, 0, 6, 2, 4, 0, 2, 0, 1, 4

Offset: 1

Reinhard Zumkeller, Oct 12 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Antti Karttunen computed from the b-file of A005179 provided by _Don Reble_)

Cf. A005179, A007949.

Cf. A099314, A099311, A099315.

A005179 = Cases[Import["https://oeis.org/A005179/b005179.txt", "Table"], {, }][[All, 2]];
a[n_] := IntegerExponent[A005179[[n]], 3];
Array[a, 2000] (* Jean-François Alcover, Dec 10 2021 *)

a(n) = A007949(A005179(n)).

More terms from Antti Karttunen, Oct 05 2017

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

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A099316 Greatest 3-smooth number dividing the n-th minimal number.

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A099312 Exponent of greatest power of 2 dividing the n-th minimal number.

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A099313 Exponent of greatest power of 3 dividing the smallest number having exactly n divisors.

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