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.

Previous Showing 11-17 of 17 results.

A072121 a(1) = 1; for n > 1, a(n) = smallest number > a(n-1) having at least n divisors.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 30, 36, 48, 60, 72, 120, 144, 168, 180, 240, 252, 336, 360, 420, 480, 504, 540, 720, 840, 900, 960, 1008, 1080, 1260, 1320, 1440, 1680, 1800, 1980, 2160, 2520, 2640, 2880, 3360, 3600, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 6300
Offset: 1

Views

Author

Vladeta Jovovic, Jun 19 2002

Keywords

Links

Crossrefs

Cf. A069654, A061799.

Programs

  • Maple
    f:= proc(n,a) local k;
for k from a+1 do
  if numtheory:-tau(k) >= n then return k fi
od
end proc:
R:= 1: x:= 1:
for n from 2 to 100 do x:= f(n,x); R:= R,x od:
R; # Robert Israel, Jan 09 2024

A103578 Number of divisors of m^2, where m is the smallest number with at least n divisors.

Original entry on oeis.org

1, 3, 5, 9, 15, 15, 21, 21, 25, 27, 45, 45, 63, 63, 63, 63, 75, 75, 81, 81, 105, 105, 105, 105, 135, 135, 135, 135, 135, 135, 189, 189, 225, 225, 225, 225, 243, 243, 243, 243, 315, 315, 315, 315, 315, 315, 315, 315, 405, 405, 405, 405, 405, 405, 405, 405, 405
Offset: 1

Views

Author

Stefan Steinerberger, Aug 31 2008

Keywords

Examples

			a(8) = 21 because smallest number with 8 divisors is 24, 24^2 = 576 and 576 has 21 divisors.

Links

Crossrefs

Cf. A000005, A061799, A141264.

Programs

  • Mathematica
    a = {}; i = 1; For[n = 1, n < 60, n++, While[DivisorSigma[0, i] < n, i++ ]; AppendTo[a, DivisorSigma[0, i^2]]]; a (* Stefan Steinerberger, Aug 31 2008 *)

Formula

a(n) = A000005((A061799(n))^2). - R. J. Mathar, Sep 01 2008

Extensions

Edited by R. J. Mathar, Sep 01 2008, Dec 15 2008
Extended beyond a(8) by R. J. Mathar, Aug 31 2008

A213918 a(n) = smallest possible element of a set of n positive integers s_1, s_2, ..., s_n such that for i != j, |s_i - s_j| = gcd(s_i, s_j), where |x| denotes absolute value.

Original entry on oeis.org

1, 1, 2, 6, 36, 210, 14976, 552720, 309582000
Offset: 1

Views

Author

Phil Scovis, Mar 04 2013

Keywords

Examples

			Examples of sets for the first few cases:
{1},
{1,2},
{2, 3, 4},
{6, 8, 9, 12},
{36, 40, 42, 45, 48},
{210, 216, 220, 224, 225, 240},
{14976, 14980, 14994, 15000, 15008, 15015, 15120},
{552720, 552825, 552960, 553000, 553014, 553140, 553280, 554400},
{309582000, 309583680, 309583800, 309583872, 309583890, 309584000, 309584025, 309584100, 309584160}.

Crossrefs

Cf. A061799, A214799.

Programs

  • Mathematica
    ok[v_, n_] := v == Select[v, GCD[#, n] == Abs[n - #] &];
ric[p_, cc_, k_] :=
If[Length@p == k, sol = p; True,
  Block[{c = cc, x, r = False},
   While[c != {}, x = First@c; c = Rest@c;
    If[p == Select[p, GCD[#, x] == Abs[x - #] &] &&
     ric[Append[p, x], c, k], r = True; Break[]]]; r]];
a[k_] := Block[{n = 1, d}, While[Length[d = Divisors@n] < k - 1 ||
!ric[{n}, n + d, k], n++]; n];
Do[Print[n, " ", a[n], " ", sol], {n, 7}]

Extensions

Corrected (with Mathematica program) by Giovanni Resta, Mar 05 2013. Entry revised by N. J. A. Sloane, Mar 05 2013
a(8) from Robert Gerbicz, Mar 05 2013
a(9) from Robert Gerbicz, Mar 06 2013

A365263 Numbers m for which A139770(m) and A140635(m) differ.

Original entry on oeis.org

16, 64, 81, 144, 192, 320, 324, 400, 448, 576, 625, 704, 729, 784, 832, 900, 960, 1024, 1088, 1216, 1296, 1344, 1458, 1472, 1600, 1728, 1764, 1856, 1936, 1984, 2025, 2112, 2240, 2304, 2368, 2401, 2496, 2500, 2624, 2704, 2752, 2880, 2916, 3008, 3072, 3136, 3264, 3392, 3520, 3600, 3645, 3648, 3776, 3904, 3969
Offset: 1

Views

Author

Hartmut F. W. Hoft, Aug 29 2023

Keywords

Links

Crossrefs

Cf. A000005, A002182, A005179, A007412, A061799, A139770, A140635.

Programs

  • Mathematica
    (* a139770[ ] and a140635[ ] are defined in their respective sequences *)
a365263[{m_, n_}] := Select[Range[m, n], a139770[#]!=a140635[#]&]
a365263[{1, 4000}]
  • PARI
    isok(m) = my(nd = numdiv(m)); for (i=1, m-1, if (numdiv(i) == nd, return (0)); if (numdiv(i)> nd, return(1))); 0; \\ Michel Marcus, Aug 31 2023

A123258 a(n) = n-th divisor of the smallest positive integer with at least n divisors.

Original entry on oeis.org

1, 2, 4, 6, 6, 12, 12, 24, 36, 48, 30, 60, 30, 40, 60, 120, 90, 180, 120, 240, 90, 120, 180, 360, 120, 144, 180, 240, 360, 720, 420, 840, 315, 420, 630, 1260, 420, 560, 840, 1680, 315, 360, 420, 504, 630, 840, 1260, 2520, 360, 420, 504, 560, 630, 720, 840, 1008
Offset: 1

Views

Author

Leroy Quet, Nov 06 2006

Keywords

Comments

a(n) = n-th divisor of A061799(n).

Examples

			The smallest positive integer with at least 11 divisors is 60, which has 12 divisors.
So a(11) = the 11th divisor of 60, which is 30.

Crossrefs

Cf. A061799.

Programs

  • Mathematica
    f[n_] := Block[{k = 1, d},While[d = Divisors[k]; Length[d] < n, k++ ];d[[n]]];Table[f[n], {n, 60}] (* Ray Chandler, Nov 11 2006 *)
Do[k = 1; While[Length[Divisors[k]] < n, k++ ]; Print[Divisors[k][[n]]], {n, 100}] (* Ryan Propper, Nov 12 2006 *)

Extensions

Extended by Ray Chandler, Nov 11 2006
More terms from Ryan Propper, Nov 12 2006

A373532 a(n) is the least number k such that A373531(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 12, 120, 240, 3276, 2520, 10920, 21840, 32760, 65520, 622440, 600600, 900900, 3636360, 1801800, 3603600, 4455360, 22407840, 8910720, 17821440, 51351300, 46060560, 69090840, 92121120, 126977760, 138181680, 380933280, 245044800, 414545040, 490089600, 507911040
Offset: 1

Views

Author

Amiram Eldar, Jun 08 2024

Keywords

Crossrefs

Cf. A061799, A328857, A373531.

Programs

  • Mathematica
    f[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[12, 10^6]
  • PARI
    f(n) = vecmax(matreduce(apply(x->eulerphi(x), divisors(n)))[ , 2]);
lista(nmax, kmax = oo) = {my(v = vector(nmax), k = 1, c = 0, i); while(c < nmax && k < kmax, i = f(k); if(i <= nmax && v[i] == 0, c++; v[i] = k); k++); v}
  • Python
    from collections import Counter
from itertools import count, islice
from sympy import divisors, totient
def agen(): # generator of terms
    adict, n = dict(), 1
    for k in count(1):
        divs = divisors(k)
        if len(divs) < n:
            continue
        c = Counter(totient(d) for d in divs)
        v = c.most_common(1)[0][1]
        if v not in adict:
            adict[v] = k
            while n in adict:
                yield adict[n]
                n += 1
print(list(islice(agen(), 11))) # Michael S. Branicky, Jun 08 2024

Formula

a(n) >= A061799(n).

A154882 a(n) is the smallest number with at least as many divisors as a(n-1)^2.

Original entry on oeis.org

2, 4, 12, 120, 7560, 8648640, 1927522396800, 4747472432036420486400, 128438082648984172330026178330296384000, 6184173455628205993842062057864303743050691444602955105860128640000
Offset: 1

Views

Author

J. Lowell, Jan 16 2009

Keywords

Examples

			4^2 = 16 has 5 divisors; the smallest number with at least 5 divisors is 12.

Formula

a(n) = A061799(A048691(a(n-1))). - R. J. Mathar, Jan 21 2009

Extensions

a(6)-a(10) from Donovan Johnson, May 09 2009
Previous Showing 11-17 of 17 results.

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