A066150 -id:A066150 - cp's OEIS frontend

A066151 Smallest n-digit number with A066150(n) divisors.

6, 60, 840, 7560, 83160, 720720, 8648640, 73513440, 735134400, 6983776800, 97772875200, 963761198400, 9316358251200, 97821761637600, 866421317361600, 8086598962041600, 74801040398884800, 897612484786617600

Offset: 1

N. J. A. Sloane, Dec 13 2001

Also largest highly composite number(A002182) with n digits. - Amiram Eldar, Jul 02 2019

			a(1) = 6 since 6 has 4 divisors and that is the record for 1-digit numbers.
a(1) = 6 since 6 has 4 divisors and k has at most 3 divisors for k < 6 and not more than 4 divisors for 6 < k <= 9.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005179, A066150. Subsequence of A002182.

a066151(m,n) = local(d,a,k,b,c); for(d=m,n,a=0; for(k=10^d,10^(d+1)-1,b=numdiv(k); if(b>a,a=b; c=k)); print1(c,","))

a(n) = A005179(A066150(n)). - Max Alekseyev, Apr 29 2010

More terms from Klaus Brockhaus, Dec 17 2001

More terms from David Wasserman, Jan 25 2002

A240544 Table (read by rows) of all k-digit positive integers (in ascending order) with maximum number of divisors A066150(k).

Original entry on oeis.org

6, 8, 60, 72, 84, 90, 96, 840, 7560, 9240, 83160, 98280, 720720, 831600, 942480, 982800, 997920, 8648640, 73513440, 82162080, 86486400, 91891800, 98017920, 99459360, 735134400, 821620800, 931170240, 994593600, 6983776800, 8454045600, 9311702400, 9448639200, 9777287520, 97772875200, 963761198400

Offset: 1

Martin Renner, Apr 07 2014

The number of elements in row k is A240543(k).

			The table T(k,m), m = 1..A240543(k), begins
6, 8;
60, 72, 84, 90, 96;
840;
etc.

Kevin P. Thompson, Table of n, a(n) for n = 1..52

Cf. A066151, A069650, A240543.

a(29)-a(35) from Giovanni Resta, Apr 08 2014

A240543 Number of n-digit positive integers with maximum number of divisors A066150(n).

Original entry on oeis.org

2, 5, 1, 2, 2, 5, 1, 6, 4, 5, 1, 1, 2, 1, 2, 3, 7, 2

Offset: 1

Martin Renner, Apr 07 2014

			a(1) = 2, since two 1-digit numbers have the maximum number of divisors 4 = #{1, 2, 3, 6} = #{1, 2, 4, 8}.
a(2) = 5, since five 2-digit numbers have the maximum number of divisors 12 = #{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} = #{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72} = #{1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84} = #{1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90} = #{1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}.

Cf. A066151, A069650, A240544.

a(10)-a(12) from Giovanni Resta, Apr 08 2014

a(13)-a(18) from Kevin P. Thompson, Sep 04 2022

A130130 a(0)=0, a(1)=1, a(n)=2 for n >= 2.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Paul Curtz, Aug 01 2007

a(n) is also total number of positive integers below 10^(n+1) requiring 9 positive cubes in their representation as sum of cubes (cf. Dickson, 1939).
A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + a(n) = A002283(n).
a(n) = number of obvious divisors of n. The obvious divisors of n are the numbers 1 and n. - Jaroslav Krizek, Mar 02 2009
Number of colors needed to paint n adjacent segments on a line. - Jaume Oliver Lafont, Mar 20 2009
a(n) = ceiling(n-th nonprimes/n) = ceiling(A018252(n)/A000027(n)) for n >= 1. Numerators of (A018252(n)/A000027(n)) in A171529(n), denominators of (A018252(n)/A000027(n)) in A171530(n). a(n) = A171624(n) + 1 for n >= 5. - Jaroslav Krizek, Dec 13 2009
a(n) is also the continued fraction for sqrt(1/2). - Enrique Pérez Herrero, Jul 12 2010
For n >= 1, a(n) = minimal number of divisors of any n-digit number. See A066150 for maximal number of divisors of any n-digit number. - Jaroslav Krizek, Jul 18 2010
Central terms in the triangle A051010. - Reinhard Zumkeller, Jun 27 2013
Decimal expansion of 11/900. - Elmo R. Oliveira, May 05 2024

Leonard Eugene Dickson, All integers except 23 and 239 are sums of eight cubes, Bulletin of the American Mathematical Society 45 (1939), p. 588-591.
Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000005, A000027, A002283, A018252, A051010, A061439, A066150, A070824, A158411, A171529, A171530, A171624, A181375, A181377, A181379, A181381, A181400, A181402, A181404.

a130130 = min 2
a130130_list = 0 : 1 : repeat 2  -- Reinhard Zumkeller, Jun 27 2013

A130130[0]:=0; A130130[1]:=1; A130130[n_]:=2; (* Enrique Pérez Herrero, Jul 12 2010 *)
A130130[n_]:=ContinuedFraction[Sqrt[1/2],n+1][[n+1]] (* Enrique Pérez Herrero, Jul 12 2010 *)
Join[{0, 1},LinearRecurrence[{1},{2},96]] (* Ray Chandler, Sep 23 2015 *)
PadRight[{0,1},120,{2}] (* Harvey P. Dale, Sep 15 2022 *)

a(n)=min(n,2) \\ Charles R Greathouse IV, Jun 01 2011

G.f.: x*(1+x)/(1-x) = x*(1-x^2)/(1-x)^2. - Jaume Oliver Lafont, Mar 20 2009
a(n) = A000005(n) - A070824(n) for n >= 1.
E.g.f.: 2*exp(x) - x - 2. - Stefano Spezia, May 19 2024

A284398 Table read by rows: T(n,k) is the number of n-digit numbers that have exactly k divisors.

Original entry on oeis.org

1, 4, 2, 2, 0, 21, 2, 30, 2, 16, 1, 10, 1, 2, 0, 5, 0, 143, 7, 260, 1, 94, 1, 170, 7, 20, 0, 92, 0, 5, 4, 47, 0, 17, 0, 11, 1, 0, 0, 16, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1061, 14, 2316, 1, 654, 0, 1934, 24, 128, 1, 943, 1, 36, 11, 753, 0, 142, 0, 146, 4, 3, 0, 433

Offset: 1

Jon E. Schoenfield, Mar 26 2017

Rows begin with row 1: [1, 4, 2, 2] for the nine 1-digit numbers 1..9 (of which one (1) has one divisor, four (the primes: 2, 3, 5, and 7) have two, two (2^2 = 4 and 3^2 = 9) have three, and two (2*3 = 6 and 2^3 = 8) have four).

The successive rows have lengths 4, 12, 32, 64, 128, 240, ... (A066150).

			Table begins:
row 1: 1, 4, 2, 2;
row 2: 0, 21, 2, 30, 2, 16, 1, 10, 1, 2, 0, 5;
row 3: 0, 143, 7, 260, 1, 94, 1, 170, 7, 20, 0, 92, 0, 5, 4, 47, 0, 17, 0, 11, 1, 0, 0, 16, 0, 0, 1, 1, 0, 1, 0, 1;
row 4: 0, 1061, 14, 2316, 1, 654, 0, 1934, 24, 128, 1, 943, 1, 36, 11, 753, 0, 142, 0, 146, 4, 3, 0, 433, 1, 0, 6, 29, 0, 43, 0, 129, 1, 0, 1, 80, 0, 0, 0, 36, 0, 7, 0, 0, 3, 0, 0, 45, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2;

Jon E. Schoenfield, Table of n, a(n) for n = 1..1696 (first 8 rows of table)

Columns k=1..6 give A000007, A006879, A379566, A379567, A379568, A379569.
Length of n-th row is A066150(n).
Cf. A000005 (number of divisors).

Table[Block[{t = KeySort[10^n - 1 + PositionIndex@ DivisorSigma[0, #] &@ Range[10^n, 10^(n + 1) - 1]]}, ReplacePart[ConstantArray[0, Max@ Keys@ t], Map[# -> Length@ Lookup[t, #] &, Keys@ t]]], {n, 0, 3}] (* Michael De Vlieger, Nov 01 2017 *)

A069650 Largest n-digit number with maximal number of divisors.

Original entry on oeis.org

8, 96, 840, 9240, 98280, 997920, 8648640, 99459360, 994593600, 9777287520, 97772875200, 963761198400, 9958865716800, 97821761637600, 978217616376000, 9651747148243200, 98930408269492800, 994651672331116800

Offset: 1

Amarnath Murthy, Apr 04 2002

Cf. A066150, A066151.

Corrected by Ray Chandler, Jun 27 2004
a(8)-a(10) from Max Alekseyev, Apr 30 2010
a(11)-a(18) from Jon E. Schoenfield, May 11 2010

cp's OEIS Frontend

A066151 Smallest n-digit number with A066150(n) divisors.

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A240544 Table (read by rows) of all k-digit positive integers (in ascending order) with maximum number of divisors A066150(k).

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A240543 Number of n-digit positive integers with maximum number of divisors A066150(n).

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A130130 a(0)=0, a(1)=1, a(n)=2 for n >= 2.

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A284398 Table read by rows: T(n,k) is the number of n-digit numbers that have exactly k divisors.

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Mathematica

A069650 Largest n-digit number with maximal number of divisors.

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