cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 17 results. Next

A350786 a(n) is the number of divisors of A061799(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 10, 12, 12, 16, 16, 16, 16, 18, 18, 20, 20, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 48, 48, 48, 48, 48, 48, 48, 48, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 64, 64, 64, 64, 72, 72, 72, 72, 72, 72, 72, 72
Offset: 1

Views

Author

J. Lowell, Jul 07 2022

Keywords

Comments

a(n) is the smallest term of A002183 that is >= n.

Examples

			a(5) = 6 because A061799(5) = 12 has 6 divisors.

Crossrefs

Cf. A000005, A061799, A002183.

Programs

  • PARI
    for(n=1, 100, my(s=n); while(numdiv(s)Michel Marcus, Jul 07 2022

Formula

a(n) = A000005(A061799(n)).

A005179 Smallest number with exactly n divisors.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144, 120, 65536, 180, 262144, 240, 576, 3072, 4194304, 360, 1296, 12288, 900, 960, 268435456, 720, 1073741824, 840, 9216, 196608, 5184, 1260, 68719476736, 786432, 36864, 1680, 1099511627776, 2880
Offset: 1

Views

Author

N. J. A. Sloane, David Singmaster

Keywords

Comments

A number n is called ordinary iff a(n)=A037019(n). Brown shows that the ordinary numbers have density 1 and all squarefree numbers are ordinary. See A072066 for the extraordinary or exceptional numbers. - M. F. Hasler, Oct 14 2014
All terms are in A025487. Therefore, a(n) is even for n > 1. - David A. Corneth, Jun 23 2017 [corrected by Charles R Greathouse IV, Jul 05 2023]

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 52.
  • Joe 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).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 89.

Links

Crossrefs

Cf. A000005, A007416, A099316, A003586, A025487, A099311, A099313, A050376, A037992, A061799, A162247, A262981, A262983.

Programs

  • Haskell
    import Data.List (elemIndex)
import Data.Maybe (fromJust)
a005179 n = succ $ fromJust $ elemIndex n $ map a000005 [1..]
-- Reinhard Zumkeller, Apr 01 2011
  • Maple
    A005179_list := proc(SearchLimit, ListLength)
local L, m, i, d; m := 1;
L := array(1..ListLength,[seq(0,i=1..ListLength)]);
for i from 1 to SearchLimit while m <= ListLength do
  d := numtheory[tau](i);
  if d <= ListLength and 0 = L[d] then L[d] := i;
  m := m + 1; fi
od:
print(L) end: A005179_list(65537,18);
# If a '0' appears in the list the search limit has to be increased. - Peter Luschny, Mar 09 2011
# alternative
# Construct list of ordered lists of factorizations of n with
# minimum divisors mind.
# Returns a list with A001055(n) entries if called with mind=2.
# Example: print(ofact(10^3,2))
ofact := proc(n,mind)
    local fcts,d,rec,r ;
    fcts := [] ;
    for d in numtheory[divisors](n) do
        if d >= mind then
            if d = n then
                fcts := [op(fcts),[n]] ;
            else
                # recursive call supposed one more factor fixed now
                rec := procname(n/d,max(d,mind)) ;
                for r in rec do
                    fcts := [op(fcts),[d,op(r)]] ;
                end do:
            end if;
        end if;
    end do:
    return fcts ;
end proc:
A005179 := proc(n)
    local Lexp,a,eList,cand,maxxrt ;
    if n = 1 then
        return 1;
    end if;
    Lexp := ofact(n,2) ;
    a := 0 ;
    for eList in Lexp do
        maxxrt := ListTools[Reverse](eList) ;
        cand := mul( ithprime(i)^ ( op(i,maxxrt)-1),i=1..nops(maxxrt)) ;
        if a =0 or cand < a then
            a := cand ;
        end if;
    end do:
    a ;
end proc:
seq(A005179(n),n=1..40) ; # R. J. Mathar, Jun 06 2024
  • Mathematica
    a = Table[ 0, {43} ]; Do[ d = Length[ Divisors[ n ]]; If[ d < 44 && a[[ d ]] == 0, a[[ d]] = n], {n, 1, 1099511627776} ]; a
(* Second program: *)
Function[s, Map[Lookup[s, #] &, Range[First@ Complement[Range@ Max@ #, #] - 1]] &@ Keys@ s]@ Map[First, KeySort@ PositionIndex@ Table[DivisorSigma[0, n], {n, 10^7}]] (* Michael De Vlieger, Dec 11 2016, Version 10 *)
mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[n] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* Vaclav Kotesovec, Apr 04 2021 *)
a[n_] := Module[{e = f[n] - 1}, Min[Times @@@ ((Prime[Range[Length[#], 1, -1]]^#) & /@ e)]]; Array[a, 100] (* Amiram Eldar, Jul 26 2025 using the function f by T. D. Noe at A162247 *)
  • PARI
    (prodR(n,maxf)=my(dfs=divisors(n),a=[],r); for(i=2,#dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a,[[n]]), r=prodR(n/dfs[i],min(dfs[i],maxf)); for(j=1,#r, a=concat(a,[concat(dfs[i],r[j])]))))); a); A005179(n)=my(pf=prodR(n,n),a=1,b); for(i=1,#pf, b=prod(j=1,length(pf[i]),prime(j)^(pf[i][j]-1)); if(bA005179(n)", ")) \\ R. J. Mathar, May 26 2008, edited by M. F. Hasler, Oct 11 2014
  • Python
    from math import prod
from sympy import isprime, divisors, prime
def A005179(n):
    def mult_factors(n):
        if isprime(n):
            return [(n,)]
        c = []
        for d in divisors(n,generator=True):
            if 1Chai Wah Wu, Aug 17 2024

Formula

a(p) = 2^(p-1) for primes p: a(A000040(n)) = A061286(n); a(p^2) = 6^(p-1) for primes p: a(A001248(n)) = A061234(n); a(p*q) = 2^(q-1)*3^(p-1) for primes p<=q: a(A001358(n)) = A096932(n); a(p*m*q) = 2^(q-1) * 3^(m-1) * 5^(p-1) for primes pA005179(A007304(n)) = A061299(n). - Reinhard Zumkeller, Jul 15 2004 [This can be continued to arbitrarily many distinct prime factors since no numbers in A072066 (called "exceptional" or "extraordinary") are squarefree. - Jianing Song, Jul 18 2025]
a(p^n) = (2*3...*p_n)^(p-1) for p > log p_n / log 2. Unpublished proof from Andrzej Schinzel. - Thomas Ordowski, Jul 22 2005
If p is a prime and n=p^k then a(p^k)=(2*3*...*s_k)^(p-1) where (s_k) is the numbers of the form q^(p^j) for every q and j>=0, according to Grost (1968), Theorem 4. For example, if p=2 then a(2^k) is the product of the first k members of the A050376 sequence: number of the form q^(2^j) for j>=0, according to Ramanujan (1915). - Thomas Ordowski, Aug 30 2005
a(2^k) = A037992(k). - Thomas Ordowski, Aug 30 2005
a(n) <= A037019(n) with equality except for n in A072066. - M. F. Hasler, Jun 15 2022

Extensions

More terms from David W. Wilson

A002183 Number of divisors of n-th highly composite number.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64, 72, 80, 84, 90, 96, 100, 108, 120, 128, 144, 160, 168, 180, 192, 200, 216, 224, 240, 256, 288, 320, 336, 360, 384, 400, 432, 448, 480, 504, 512, 576, 600, 640, 672, 720, 768, 800, 864, 896
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Record values of tau(n).
RECORDS transform of A000005.
All powers of 2 are present through 2^17. No power of 2 above that is present at least through 2^51. - Comment from Robert G. Wilson v, modified by Ray Chandler, Nov 10 2005
No power of 2 above 2^17 is contained in this sequence - see McRae link for proof. - Graeme McRae, Apr 27 2006
All numbers of the form 9*2^n are present for n=0 through n=30. - Richard Peterson, Sep 07 2024

References

  • S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. 87.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000005, A002182, A002201, A006218, A061799, A070319, A243220, A261100, A329605, A329902.

Programs

  • Haskell
    import Data.List (nub)
a002183 n = a002183_list !! (n-1)
a002183_list = nub $ map (a000005 . a061799) [1..]
-- Reinhard Zumkeller, Apr 01 2011
  • Mathematica
    Reap[ For[ record = 0; n = 1, n <= 10^9, n = If[n < 60, n+1, n+60], tau = DivisorSigma[0, n]; If[tau > record, record = tau; Print[tau]; Sow[tau]]]][[2, 1]] (* Jean-François Alcover, Aug 13 2013 *)
DeleteDuplicates[DivisorSigma[0,Range[3*10^6]],GreaterEqual] (* The program generates the first 42 terms of the sequence. *) (* Harvey P. Dale, Aug 12 2025 *)

Formula

a(n) = A000005(A002182(n)).
Also record values of differences A006218(p)-A006218(p-1). These record values occur for any p = A002182(q) where q>=2. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(A261100(n)) = A070319(n). - Antti Karttunen, Jun 06 2017
a(n) = A329605(A329902(n)). - Antti Karttunen, Jan 14 2020

Extensions

More terms from Robert G. Wilson v, Jul 24 2002

A096179 Triangle read by rows: T(n,k) is the smallest positive integer having at least k of the first n positive integers as divisors.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 4, 12, 1, 2, 4, 12, 60, 1, 2, 4, 6, 12, 60, 1, 2, 4, 6, 12, 60, 420, 1, 2, 4, 6, 12, 24, 120, 840, 1, 2, 4, 6, 12, 24, 72, 360, 2520, 1, 2, 4, 6, 12, 24, 60, 120, 360, 2520, 1, 2, 4, 6, 12, 24, 60, 120, 360, 2520, 27720, 1, 2, 4, 6, 12, 12, 24, 60, 120, 360
Offset: 1

Views

Author

Matthew Vandermast, Jun 19 2004

Keywords

Examples

			Triangle begins:
1
1 2
1 2 6
1 2 4 12
1 2 4 12 60
1 2 4  6 12 60

Links

Crossrefs

Main diagonal is A003418. Minimum in column k is A061799(k). See also A094348, A096180.

Programs

  • Maple
    with(combstruct):
a096179_row := proc(n) local k,L,l,R,LCM,comb;
R := NULL; LCM := ilcm(seq(i,i=[$1..n]));
for k from 1 to n-1 do
   L := LCM;
   comb := iterstructs(Combination(n),size=k):
   while not finished(comb) do
      l := nextstruct(comb);
      L := min(L,ilcm(op(l)));
   od;
   R := R,L;
od;
R,LCM end; # Peter Luschny, Dec 06 2010
  • Mathematica
    (* Triangular *)
A096179[n_,k_]:=Min[LCM@@@Subsets[Range[n],{k}]];
A002024[n_]:=Floor[1/2+Sqrt[2*n]];
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
(* Linear *)
A096179[n_]:=A096179[n]=A096179[A002024[n],A002260[n]];
(* Enrique Pérez Herrero_, Dec 08 2010 *)
  • PARI
    A096179(n,k)={ my(m=lcm(vector(k,i,i))); forvec(v=vector(k-1,i,[2,n]), m>lcm(v) & m=lcm(v), 2); m } \\ M. F. Hasler, Nov 30 2010

Formula

T(n,k) = min { lcm(x_1,...,x_k) ; 0 < x_1 < ... < x_k <= n }

A139770 Smallest number having at least as many divisors as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 6, 4, 6, 2, 12, 2, 6, 6, 12, 2, 12, 2, 12, 6, 6, 2, 24, 4, 6, 6, 12, 2, 24, 2, 12, 6, 6, 6, 36, 2, 6, 6, 24, 2, 24, 2, 12, 12, 6, 2, 48, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 60, 2, 6, 12, 24, 6, 24, 2, 12, 6, 24, 2, 60, 2, 6, 12, 12, 6, 24, 2, 48, 12, 6, 2, 60, 6, 6, 6, 24, 2
Offset: 1

Views

Author

J. Lowell, May 20 2008

Keywords

Comments

Similar to A140635, except that a(n) is allowed to have more divisors than n.
a(n) <= n for all n. Moreover, a(n) = n if and only if n belongs to A061799 (or equivalently A002182).
When n is prime, a(n) = 2. - Michel Marcus, Jun 14 2013
For numbers k such that a(k) and A140635(k) are not equal see A365263. - Michel Marcus, Aug 31 2023

Examples

			16 has 5 divisors; smallest number with at least 5 divisors is 12 with 6 divisors, thus a(16) = 12.

Links

Crossrefs

Cf. A000005, A002182, A061799, A140635, A365263.

Programs

  • Mathematica
    a139770[n_] := NestWhile[#+1&, 1, DivisorSigma[0, n]>DivisorSigma[0, #]&]
a139770[{m_, n_}] := Map[a139770, Range[m, n]]
a139770[{1, 89}] (* Hartmut F. W. Hoft, Jun 13 2023 *)
  • PARI
    a(n) = {nd = numdiv(n); for (i=1, n-1, if (numdiv(i) >= nd, return (i));); return (n);} \\ Michel Marcus, Jun 14 2013
  • Python
    from sympy import divisor_count as d
def a(n):
    x=d(n)
    m=1
    while True:
        if d(m)>=x: return m
        else: m+=1 # Indranil Ghosh, May 27 2017

Formula

a(n) = A061799(A000005(n)).

Extensions

Edited and extended by Ray Chandler, May 24 2008

A058198 Where d(m) (number of divisors, A000005) has risen by at least n.

Original entry on oeis.org

2, 6, 12, 12, 24, 24, 48, 48, 60, 60, 120, 120, 168, 168, 180, 180, 240, 240, 360, 360, 360, 360, 720, 720, 720, 720, 720, 720, 840, 840, 1260, 1260, 1260, 1260, 1680, 1680, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 3360, 3360, 5040, 5040, 5040, 5040
Offset: 1

Views

Author

N. J. A. Sloane, Nov 28 2000

Keywords

Comments

a(n) exists for all n (Turán, 1954). - Amiram Eldar, Apr 13 2024
a(n) >= A061799(n). - David A. Corneth, Apr 13 2024

Examples

			d(11) = 2, d(12) = 6 gives first jump of >= 3, so a(3) = a(4) = 12.

References

  • József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, p. 39, section II.1.3.a.

Links

Crossrefs

Equals A058197(n) + 1.
Cf. A000005, A051950, A058199, A061799.

Programs

Extensions

More terms from James Sellers, Nov 29 2000

A086921 Least number with at least n divisors that are at most its square root.

Original entry on oeis.org

1, 4, 12, 24, 36, 60, 120, 120, 180, 240, 360, 360, 720, 720, 720, 840, 1260, 1260, 1680, 1680, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040, 5040, 5040, 7560, 7560, 10080, 10080, 10080, 10080, 15120, 15120, 15120, 15120, 20160, 20160, 25200, 25200
Offset: 1

Views

Author

N. J. A. Sloane, Sep 19 2003

Keywords

Comments

Smallest number m such that can write m = p*q (with p <= q) in at least n ways.

Crossrefs

Cf. A038548, A038549, A004778.
Cf. A118224.

Formula

a(n) = A061799(2n-1). - David Wasserman, Apr 12 2005

Extensions

More terms from Vladeta Jovovic and Ray Chandler, Sep 20 2003

A214799 Let S be a set of n positive numbers such that all n choose 2 pairwise GCD's are distinct, and let max(S) denote the largest element of S; a(n) is the minimal value of max(S) over all choices for S.

Original entry on oeis.org

1, 2, 6, 18, 54, 120, 240, 480, 960, 1920
Offset: 1

Views

Author

N. J. A. Sloane, Mar 07 2013

Keywords

Examples

			n   a(n)      Example of S
---------------------------
1    1       {1}
2    2       {1,2}
3    6       {2,3,6}
4   18       {4,9,12,18}
5   54       {8,24,27,36,54}
6  120       {45,80,84,90,112,120}
7  240       {45,126,160,168,180,224,240}
8  480       {135,252,270,320,336,360,448,480}
9  960       {504, 640, 672, 720, 756, 810, 896, 945, 960}
10 1920      {1008, 1215, 1280, 1344, 1440, 1512, 1620, 1792, 1890, 1920}

Crossrefs

Cf. A061799 (minimum of smallest element), A213918.

Formula

Conjecture: a(n) = 15 * 2^(n-3) for n >= 6 (Robert Israel)

Extensions

a(4)-a(8) from Robert Israel, Mar 05 2013.
a(9)-a(10) from Giovanni Resta, Mar 06 2013

A118224 Smallest number having at least 2*n divisors.

Original entry on oeis.org

2, 6, 12, 24, 48, 60, 120, 120, 180, 240, 360, 360, 720, 720, 720, 840, 1260, 1260, 1680, 1680, 2520, 2520, 2520, 2520, 5040, 5040, 5040, 5040, 5040, 5040, 7560, 7560, 10080, 10080, 10080, 10080, 15120, 15120, 15120, 15120, 20160, 20160, 25200, 25200, 25200
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 16 2006

Keywords

Links

Crossrefs

Cf. A086921, A003680 (exactly 2n).

Programs

  • Mathematica
    s={};Do[i=1;Until[DivisorSigma[0,i]>=2n,i++];AppendTo[s,i],{n,45}];s (* James C. McMahon, Sep 24 2024 *)
  • Python
    from sympy import divisors
def a(n):
  m = 4*n - 2
  while len(divisors(m)) < 2*n: m += 1
  return m
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Feb 06 2021

Formula

A000005(a(n)) >= 2*n;
a(n) = A061799(2*n).
a(n) <= A003680(n). - Michael S. Branicky, Feb 06 2021

A347064 Smallest number with at least 2^n divisors.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 17297280, 294053760, 5587021440, 128501493120, 3212537328000, 93163582512000, 2888071057872000, 106858629141264000, 4381203794791824000, 184010559381256608000, 7912454053394034144000, 371885340509519604768000
Offset: 0

Views

Author

Jon E. Schoenfield, Aug 15 2021

Keywords

Comments

Begins to differ from A037992 at n=18; a(18) < A037992(18), but the number of divisors d(a(18)) = 276480 > 262144 = 2^18.

Examples

			   n                A037992(n)                      a(n)  d(a(n))      2^n
  --  ------------------------  ------------------------  -------  -------
  18     188391763176048432000     184010559381256608000   276480   262144
  19    8854412869274276304000    7912454053394034144000   552960   524288
  20  433866230594439538896000  371885340509519604768000  1105920  1048576

Links

Crossrefs

Cf. A000005, A000079, A037992, A061799.

Programs

  • Mathematica
    Table[SelectFirst[Table[{n,DivisorSigma[0,n]},{n,0,11*10^5}],#[[2]]==2^k&],{k,0,8}][[;;,1]] (* The program generates the first nine terms of the sequence. *)  (* Harvey P. Dale, Feb 04 2024 *)

Formula

a(n) = A061799(2^n). - Michel Marcus, Aug 16 2021
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