A140635 -id:A140635 - cp's OEIS frontend

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

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

Hartmut F. W. Hoft, Aug 29 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

(* a139770[ ] and a140635[ ] are defined in their respective sequences *)
a365263[{m_, n_}] := Select[Range[m, n], a139770[#]!=a140635[#]&]
a365263[{1, 4000}]

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

A306585 Start with n and find the LCM of n and A140635(n), and continue until a number m is reached such that A140635(m) = m.

Original entry on oeis.org

1, 2, 6, 4, 120, 6, 840, 24, 36, 120, 83160, 12, 1081080, 840, 120, 16, 294053760, 36, 5587021440, 60, 840, 83160, 128501493120, 24, 900, 1081080, 7560, 2520, 93163582512000, 120, 2888071057872000, 10080, 83160, 294053760, 840, 36, 106858629141264000, 5587021440, 1081080, 120, 4381203794791824000

Offset: 1

J. Lowell, Feb 25 2019

nonn

a(n) = n is a subsequence of minimal numbers A007416, i.e., when A140635(n) = n. It appears that p_n > log_2(sigma_0(a(p_n))) for all primes p_n, and that a(p_n) form an increasing subsequence satisfying a(k) < a(p_n) for all k < p_n. - Hartmut F. W. Hoft, Mar 14 2023

			a(5) = 120 because:
A140635(5) = 2 and LCM of 5 and 2 is 10,
A140635(10) = 6 and LCM of 10 and 6 is 30,
A140635(30) = 24 and LCM of 30 and 24 is 120,
A140635(120) = 120 so a(5) = 120.
From _Hartmut F. W. Hoft_, Mar 14 2023: (Start)
Table of iteration steps starting at n and ending with fixed point a(n):
    1  ...
    2  ...
    3    6   ...
    4  ...
    5   10    30   120    ...
    6    6   ...
    7   14    42   168    840     ...
    8   24   ...
    9   36   ...
   10   30   120   ...
   11   22    66   264   1320    9240   83160       ...
   12  ...
   13   26    78   312   1560   10920   98280   1081080  ...
   14   42   168   840    ...
   15   30   120   ...
   16  ...
  ... (End)

Hartmut F. W. Hoft, Table of n, a(n) for n = 1..135

Cf. A005179, A007416, A140635.

(* a005179[ ] based on the function by Vaclav Kotesovec in A005179 *)
mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[mHartmut F. W. Hoft, Mar 14 2023 *)

s(n) = my(nd=numdiv(n)); for(k=1, n, if(numdiv(k) == nd, return(k))); \\ A140635
a(n) = my(m=n, t=s(n)); while(1, m=lcm(m, t); t=s(m); if(m==t, return(m))); \\ Daniel Suteu, Feb 25 2019

a(13)-a(37) from Rémy Sigrist, Feb 25 2019

Terms a(38) and beyond from Hartmut F. W. Hoft, Mar 14 2023

A351162 a(1)=2048. For n>1, a(n) is the LCM of a(n-1) and A140635(a(n-1)).

Original entry on oeis.org

2048, 30720, 645120, 7096320, 92252160, 1383782400, 23524300800, 446961715200, 10280119449600, 71960836147200, 2086864248268800, 64692791696332800, 582235125266995200, 21542699634878822400, 883250685030031718400, 37979779456291363891200, 189898897281456819456000

Offset: 1

J. Lowell, Feb 04 2022

nonn

Similar sequences starting with smaller powers of 2 are known to converge after a few terms.

This sequence is constant from n = 35. I.e.: a(n) = a(35) for all n >= 36. - Daniel Suteu, Mar 15 2022

			2048 has 12 divisors. LCM of 2048 and 60 (smallest number with 12 divisors) is 30720.

Daniel Suteu, Table of n, a(n) for n = 1..36
Daniel Suteu, Perl program

Cf. A140635.

f[n_] := Module[{d = DivisorSigma[0, n], k = 1}, While[DivisorSigma[0, k] != d, k++]; k]; a[1] = 2048; a[n_] := a[n] = LCM[a[n - 1], f[a[n - 1]]]; Array[a, 5] (* Amiram Eldar, Feb 04 2022 *)

a(5)-a(7) from Amiram Eldar, Feb 04 2022

a(8)-a(17) from Jon E. Schoenfield and Daniel Suteu, Mar 15 2022

A019505 a(1)=1; for n > 1, a(n) is the smallest number with the same number of divisors as 2*a(n-1).

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 720, 1260, 2520, 5040, 10080, 20160, 27720, 55440, 110880, 221760, 332640, 665280, 1081080, 2162160, 4324320, 8648640, 17297280, 21621600, 43243200, 73513440, 147026880, 294053760, 367567200, 735134400, 1396755360, 2793510720

Offset: 1

J. Lowell

nonn

From J. Lowell, Mar 19 2012 and Apr 05 2012: (Start)
Conjectures:
Subsequence of A002182. [This conjecture is false. The 64th term is 97039187544499200, which has exactly 63360 divisors, but is NOT in A002182; which has the smaller number 74801040398884800, which has 64512 divisors. - J. Lowell, Nov 29 2021]
In order for n to be followed by a number less than 2n, a requirement is that the number of 2's in the prime factorization of n must not be of the form p-2 where p is a prime.
There are infinitely many values where n, 2n, and 3n are all in this sequence. (It can be proved that n, 2n, 3n, and 4n can never all be in this sequence.)
In any group of 3 consecutive terms of this sequence a,b,c at most one of the following statements is true:
The value of b is less than twice a.
The value of c is less than twice b. [This conjecture is false. Terms 121-123 are 9363553722094352358689983872000, 14258138622280036546187020896000, and 26139920807513400334676204976000. - J. Lowell, Jan 28 2022]
There are terms divisible by 2^k no matter how large k is.
The primes and the powers of 3 are the only numbers that never "come and go" from the lists of divisors of the numbers in the sequence as the terms advance. [This conjecture is false. The 122nd term, 14258138622280036546187020896000, is a multiple of 3^5=243, but the 123rd term, 26139920807513400334676204976000, is not. - J. Lowell, Jan 28 2022]
(End)

			After a(3)=4 we argue as follows: 2*4 = 8 has 4 factors (1,2,4,8), but smallest number with 4 factors is 6, so a(4)=6.

David A. Corneth, Table of n, a(n) for n = 1..172 (first 63 terms from R. J. Mathar)
Michael De Vlieger, Concordance of a(n), A002182, and A025487, n = 1..176, giving indices in A002182 and A025487 and number of divisors of a(n).

Cf. A020697, A140635.

a(n) = A140635(2*a(n-1)). - J. Lowell, May 20 2008

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

J. Lowell, May 20 2008

nonn

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

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

Antti Karttunen, Table of n, a(n) for n = 1..10000

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

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 *)

a(n) = {nd = numdiv(n); for (i=1, n-1, if (numdiv(i) >= nd, return (i));); return (n);} \\ Michel Marcus, Jun 14 2013

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

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

Edited and extended by Ray Chandler, May 24 2008

A061300 Least number whose number of divisors is n!.

Original entry on oeis.org

1, 1, 2, 12, 360, 55440, 61261200, 293318625600, 6064949221531200, 1315675499575984747200, 1130066578473302698988760000, 8029566026151577210973143393920000, 44532446925432190155112500678140561280000, 89867631285897528426742043782255216503577152000000

Offset: 0

Amarnath Murthy and Labos Elemer, Apr 26 2001

a(n) = A037019(n!) for all n <= 12 except for 4. I conjecture that this remains true for all larger n, i.e., 4! is the only "exceptional" factorial (see A037019). - David Wasserman, Jun 13 2002
Conjecture is confirmed for n <= 30. - Max Alekseyev, Sep 05 2023
Alternate definition: a(0)=1; for n >= 1, smallest number with same number of divisors as A006939(n-1). - J. Lowell, May 20 2008

			a(3) = 12 and tau(12) = 6 = 3!.

Max Alekseyev, Table of n, a(n) for n = 0..30

Cf. A000005, A005179, A007304, A006939, A037019, A000142, A072066, A009287.

Cf. A140635.

a(n) = A005179(n!); for example, A005179(120)=55440.

a(n) = Min{x| A000005(x)=n!}; for example, A000005(55440)=120 and 55440 is minimal.

More terms from David Wasserman, Jun 13 2002

Terms a(12) onward from Max Alekseyev, Sep 05 2023

A134865 Numbers k meeting the following criterion: if k is a multiple of d, then it is also a multiple of the smallest number with same number of divisors as d.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 120, 240, 360, 720, 2520, 5040, 7560, 10080, 15120, 20160, 45360, 50400, 100800, 332640, 352800, 665280, 705600, 4324320, 8648640, 17297280, 21621600, 43243200, 13492656777600

Offset: 1

J. Lowell, Jan 29 2008

Note that this is not a subsequence of A002182: 100800 is in this sequence but not in A002182. - J. Lowell, Feb 22 2008
A subset of A005179. - Max Alekseyev, May 19 2008
A number k is in this sequence iff for every divisor d of k, A005179(A000005(d)) (= A140635(d)) is also a divisor of k. So the question of the finiteness of this sequence is closely related to the form of the elements of A005179. - Max Alekseyev, May 19 2008, May 20 2008
Rearrangement of this sequence, forming a subsequence of A005179, is given by A140753. Corresponding indices of elements of A005179 are given by A138394 and A140752. - Max Alekseyev, May 26 2008
A subsequence of A007416 which is a subsequence of A025487, so every term is primally tight and even (after the first term). Thus if d is a divisor of a term, then the least integer with the same prime signature as d (=A046523(d)) is also a divisor. So only the divisors that are in A025487 need be tested. - Ray Chandler
a(32) > 8*10^25 if it exists. - David A. Corneth, Dec 10 2021

			60 is a multiple of 30 with 8 divisors, but not of 24 (the smallest number with 8 divisors) so 60 is not a term of this sequence.

Cf. A000005, A005179, A007416, A138394, A140752, A140753.

a = {}; For[n = 1, n < 10000, n++, b = Divisors[n]; c = 1; For[i = 1, i < Length[b] + 1, i++, j = 1; While[Length[Divisors[j]] < Length[Divisors[b[[i]]]], j++ ]; If[ ! Mod[n, j] == 0, c = 0]]; If[c == 1, AppendTo[a, n]]]; a (* Stefan Steinerberger, Feb 05 2008 *)

isA134865(n)={ n%2 & return(n==1); fordiv(n, d, bigomega(d)>1 || next; nd=numdiv(d); for(k=4, d, numdiv(k)==nd || next; n%k & return; break)); 1 }
for(n=1,10^7,if(isA134865(n),print1(n,", "))); \\ R. J. Mathar, May 17 2008

a(n) = A005179(A140752(n)). - Max Alekseyev, May 26 2008

More terms from Stefan Steinerberger, Feb 05 2008
More terms from J. Lowell, Feb 22 2008
a(22)-a(30) from Don Reble, May 17 2008
a(31)=13492656777600 from Ray Chandler, Jun 30 2008

A069805 Smallest k>n such that tau(n) divides tau(k)=A000005(k).

Original entry on oeis.org

2, 3, 5, 9, 6, 8, 8, 10, 12, 14, 12, 18, 14, 15, 21, 48, 18, 20, 20, 28, 22, 24, 24, 30, 28, 27, 30, 32, 30, 40, 32, 44, 34, 35, 38, 100, 38, 39, 40, 42, 42, 54, 44, 45, 50, 51, 48, 80, 50, 52, 54, 60, 54, 56, 56, 66, 58, 60, 60, 72, 62, 65, 68, 192, 66, 70, 68, 72, 70, 78, 72

Offset: 1

Benoit Cloitre, May 01 2002

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000005.

Cf. A140635 (without k>n constraint).

for(s=1,100,n=s+1; while(frac(numdiv(n)/numdiv(s))>0,n++); print1(n,","); )

a(n) = my(k=n+1); while(numdiv(k)%numdiv(n), k++); k; \\ Michel Marcus, Sep 15 2018

A138113 a(n) is the smallest number with same number of divisors as n*a(n-1).

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 27720, 166320, 1441440, 7207200, 73513440, 367567200, 4655851200, 64250746560, 1028011944960, 5140059724800, 32607253879200, 260858031033600, 4620913692595200, 97039187544499200, 897612484786617600, 9873737332652793600

Offset: 1

J. Lowell, May 04 2008

nonn

			For n=8, n*a(n-1) = 8*5040 = 40320 has 96 divisors, but the smallest number with 96 divisors is 27720, so a(8)=27720.

Cf. A019505 (2*a(n-1)).

Cf. A140635.

A138113(n)={ local(an1,t) ; if(n<=2, return(n) ) ; an1 = A138113(n-1) ; t=length(divisors(n*an1)) ; return(A005179(t)) ; } {for (n=1,40, print1(A138113(n)", ") ; ) } \\ R. J. Mathar, Mar 20 2010

a(n) = A140635(n*a(n-1)). - J. Lowell, May 20 2008

More terms from R. J. Mathar, Mar 20 2010

a(23) from Jon E. Schoenfield, Mar 17 2022

A347982 a(n) is the greatest k, 0 < k < n, such that tau(k) = tau(n), or -1 if no such k exists, where tau is A000005.

Original entry on oeis.org

-1, -1, 2, -1, 3, -1, 5, 6, 4, 8, 7, -1, 11, 10, 14, -1, 13, 12, 17, 18, 15, 21, 19, -1, 9, 22, 26, 20, 23, 24, 29, 28, 27, 33, 34, -1, 31, 35, 38, 30, 37, 40, 41, 32, 44, 39, 43, -1, 25, 45, 46, 50, 47, 42, 51, 54, 55, 57, 53, -1, 59, 58, 52, -1, 62, 56, 61, 63, 65, 66, 67

Offset: 1

David James Sycamore, Sep 22 2021

sign

a(n) = -1 if and only if n is a term in A005179.

			a(1) = -1 because there is no positive number less than 1 having 1 divisor.
a(2) = -1 because 2 is the first prime.
a(3) = 2 because 2 is the greatest prime less than 3 and all primes have 2 divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A005179, A079427, A079427, A140635.

a[1] = -1; a[n_] := Module[{k = n - 1, d = DivisorSigma[0, n]}, While[k > 0 && DivisorSigma[0, k] != d, k--]; If[k == 0, -1, k]]; Array[a, 100] (* Amiram Eldar, Sep 23 2021 *)

a(n) = my(nd=numdiv(n)); forstep(k=n-1, 1, -1, if (numdiv(k)==nd, return(k))); return(-1); \\ Michel Marcus, Sep 22 2021

cp's OEIS Frontend

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

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A306585 Start with n and find the LCM of n and A140635(n), and continue until a number m is reached such that A140635(m) = m.

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A351162 a(1)=2048. For n>1, a(n) is the LCM of a(n-1) and A140635(a(n-1)).

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A019505 a(1)=1; for n > 1, a(n) is the smallest number with the same number of divisors as 2*a(n-1).

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

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A061300 Least number whose number of divisors is n!.

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A134865 Numbers k meeting the following criterion: if k is a multiple of d, then it is also a multiple of the smallest number with same number of divisors as d.

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A069805 Smallest k>n such that tau(n) divides tau(k)=A000005(k).

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