A007416 - OEIS

A007416

The minimal numbers: sequenceA005179arranged 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers k such that there is no x < k such thatA000005(x) =A000005(k). -Benoit Cloitre,Apr 28 2002

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

Subsequence ofA025487.If some m inA025487is 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 ofA025487;A002182is a subsequence.

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

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

Cf.A053212andA064787(the sequence {A000005(a(n))} and its inverse permutation).