A002183 -id:A002183 - cp's OEIS frontend

A095717 "Second order" highly composite numbers: the gap between the number of divisors (d(n)) rises to a new record.

2, 12, 120, 720, 2520, 5040, 110880, 1441440, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 481880599200, 963761198400, 6746328388800, 55898149507200, 130429015516800, 195643523275200, 1732842634723200, 4043299481020800, 6064949221531200, 60649492215312000

Offset: 1

Stefano Lanfranco (lastefano(AT)yahoo.it), Jul 08 2004

The corresponding indices of the highly composite numbers are 2, 5, 10, 14, 18, 19, 30, 40, ... (see the link for more values). - Amiram Eldar, Jul 17 2019

			120 is in the sequence because d(120)=16 and the previous highly composite number is 60 with d(60)=12, the gap between the number of divisor 16-12=4 is the maximum with number <=120

Amiram Eldar, Table of n, a(n) for n = 1..1374
Amiram Eldar, Table of n and k such that A002182(k) = a(n) for n = 1..1374

Cf. A002182, A002183, A053640, A053624.

s={}; dmax = dmprev= gapmax=0; Do[d = DivisorSigma[0, k]; If[d > dmax ,  dmprev = dmax; dmax = d; gap = dmax - dmprev ;If[gap > gapmax, gapmax = gap; AppendTo[s, k]]], {k, 1, 1500000}]; s (* Amiram Eldar, Jul 17 2019 *)

Definition edited by Harvey P. Dale, Apr 09 2018

More terms from Amiram Eldar, Jul 17 2019

A141320 Both n and the smallest number with n divisors are in A002182.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 1680, 5040, 10080, 20160

Offset: 1

J. Lowell, Aug 02 2008

Question: is this sequence finite? See A189394 for detailed information.

Intersection of A002182 and A002183. - Jianing Song, Apr 03 2018

			Both 20160 and the smallest number with 20160 divisors, 195643523275200, are in A002182, so 20160 is a term.

Cf. A002182, A002183, A189394.

a(n) = tau(A189394(n)) = A000005(A189394(n)). - Jianing Song, Apr 03 2018

a(16)-a(18) from Jianing Song, Apr 03 2018

Keywords fini, full are copied over from A189394 by Max Alekseyev, Feb 02 2025

A161887 A product of quotients of factorials.

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 840, 7560, 15120, 110880, 166320, 1441440, 2882880, 10810800, 43243200, 183783600, 367567200, 2793510720, 6983776800, 58663725120, 117327450240, 299836817280, 2698531355520, 7495920432000

Offset: 1

Peter Luschny, Jun 21 2009

Definition: Let b(n,k) = floor(n/2^k)! and m = log[2](n) then c(n) = product_{k=1..m} b(n,k) / b(n,k+1)^2.
a(n) is the sequence derived from c(n) by eliminating duplicates and sorting the values.
a(1) through a(19) are highly composite numbers (A002182).
The number of divisors of a(1) through a(28) are number of divisors of highly composite numbers (A002183).
A055773(floor(n/2)) is a divisor of a(n), a(n)/A055773(floor(n/2)) after eliminating duplicates and sorting starts 1,4,24,216,1440,2160,..

Amiram Eldar, Table of n, a(n) for n = 1..1669

Cf. A002182, A002183.

a := proc(n) local m,k; m := nops(convert(n,base,2));
mul(iquo(n,2^k)!/iquo(n,2^(k+1))!^2,k=1..m-1) end:
seq(a(i),i=1..50): A:=sort(convert({%},list));

b[n_, k_] := Floor[n/2^k]!; c[n_] := Product[b[n, k]/b[n, k+1]^2, {k, 1, Log[2, n]}]; A = Array[c, 50] // DeleteDuplicates // Sort (* Jean-François Alcover, Jun 17 2019 *)

A243423 Number of divisors of A243220(n).

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 2, 3, 4, 2, 4, 4, 4, 4, 4, 3, 4, 4, 6, 2, 6, 4, 6, 6, 6, 3, 6, 5, 4, 4, 6, 5, 6, 4, 6, 5, 6, 6, 3, 6, 4, 6, 8, 5, 4, 6, 4, 8, 8, 4, 4, 8, 5, 8, 8, 6, 6, 8, 5, 6, 8, 2, 8, 8, 6, 6, 9, 6, 4, 8, 6, 6, 6, 9, 6, 4, 9, 6, 8, 4, 10, 6, 6, 6, 9, 8, 2, 10, 8, 8, 12, 9, 10

Offset: 1

Alexander R. Povolotsky, Jun 04 2014

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002183, A243220.

A342833 Integers m such that the number of divisors whose last digit equals the last digit of m sets a new record.

Original entry on oeis.org

1, 11, 40, 60, 120, 240, 360, 480, 600, 1200, 1800, 2400, 3600, 7200, 8400, 12600, 16800, 25200, 50400, 75600, 100800, 151200, 201600, 252000, 277200, 453600, 504000, 554400, 831600, 1108800, 1663200, 2217600, 2772000, 3326400, 4989600, 5544000, 6652800, 7207200

Offset: 1

Bernard Schott, Mar 23 2021

Inspired by Project Euler, Problem 474 (see link).

The corresponding number of divisors whose last digit equals the last digit: 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, ...

			a(5) = 120 is in the sequence because A330348(120) = 6, the six corresponding divisors are {10, 20, 30, 40, 60, 120} and 6 is larger than any earlier value in A330348.

Project Euler, Problem 474: Last digits of divisors.

Cf. A002182, A002183, A330348, A335491.

d[n_] := DivisorSum[n, 1 &, Mod[# - n, 10] == 0 &]; dm = 0; s = {}; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 10^7}]; s (* Amiram Eldar, Mar 23 2021 *)

f(n) = my(dig = n%10); sumdiv(n, d, d%10 == dig); \\ A330348
lista(nn) = my(m, k=0, kk); for (n=1, nn, kk = f(n); if (kk>k, print1(n, ", "); k = kk)); \\ Michel Marcus, Mar 24 2021

For n >= 3, a(n) = 10 * A002182(n) (conjectured).

A346729 Maximum number of divisors among n-bit numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 120, 144, 168, 200, 240, 288, 360, 432, 504, 600, 720, 864, 1008, 1152, 1344, 1600, 1920, 2304, 2688, 3072, 3584, 4096, 4800, 5760, 6720, 7680, 8640, 10080, 11520, 13824, 16128, 18432, 20736, 23040, 27648

Offset: 1

Jon E. Schoenfield, Jul 30 2021

nonn

a(n) is the maximum value of tau(k)=A000005(k) for k in the interval [2^(n-1), 2^n - 1]. For n >= 3, that smallest k at which tau(k) is maximized in that interval is A036484(n).

No term is repeated: for n >= 1, if k is the number in [2^(n-1), 2^n - 1] at which tau(k) is maximized (i.e., tau(k) = a(n)), then 2k, which will be a number in [2^n, 2^(n+1) - 1], will have more divisors than k has, so a(n+1) >= tau(2k) > tau(k) = a(n).

			There are four 3-bit numbers: 4 = 100_2, 5 = 101_2 = 5, 6 = 110_2, 7 = 111_2. 5 and 7 are both prime, so each has 2 divisors; 4 = 2^2 has 3 divisors (1, 2, and 4), and 6 = 2*3 has 4 divisors (1, 2, 3, and 6). Thus, the maximum number of divisors among 3-bit numbers is A000005(6) = 4, so a(3)=4.

Cf. A000005, A002183, A036484, A346730.

a[n_]:=Max[Table[DivisorSigma[0,k],{k,2^(n-1),2^n-1}]]; Table[a[n],{n,23}] (* Stefano Spezia, Aug 02 2021 *)

a(n) = vecmax(apply(numdiv, [2^(n-1)..2^n-1])); \\ Michel Marcus, Aug 03 2021

from sympy import divisors
def a(n): return max(len(divisors(n)) for n in range(2**(n-1), 2**n))
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Aug 02 2021

A356078 Highly composite numbers that are multiples of their number of divisors.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 5040, 10080, 15120, 20160, 25200, 55440, 110880, 221760, 277200, 665280, 720720, 1441440, 2882880, 3603600, 8648640, 14414400, 32432400, 43243200, 61261200, 245044800, 551350800, 735134400, 1102701600, 2205403200

Offset: 1

J. Lowell, Jul 25 2022

nonn

Conjecture: this sequence is finite. The prime factorization of the number of divisors of a large highly composite number usually has a high power of 2; this is not true of the number of divisors of a highly composite number itself.

			180 (in A002182) has 18 divisors, and 180/18 equals 10; so, 180 is in this sequence.

Intersection of A002182 and A033950.

Cf. A002183.

More terms from Amiram Eldar, Jul 25 2022

A175189 Smallest integer m such that phi(phi(m))^n + tau(phi(m))^n = phi(rad(m))^n, where n is the number of iterations of phi(phi), tau(phi) and phi(rad) functions.

Original entry on oeis.org

7, 33, 29, 59, 347, 2039, 4079, 32633, 65267, 913739, 1827479, 36549581

Offset: 1

Michel Lagneau, Mar 01 2010

Remarks about an interesting property of the equation phi(phi(m))^k + tau(phi(m))^k = phi(rad(m))^k: Let p be a prime number. If p is a solution of this equation with k iterations, and if q = 2*p+1 is prime, then q is solution of the equation with k+1 iterations.
Proof: we use the following properties, if p is prime: phi(phi(2*p+1)) = phi(2*p) = p-1; tau(phi(2*p+1)) = tau(2*p) = 4; phi(rad(2*p+1)) = phi(2*p+1) = 2*p; phi(phi(p)) = phi(p-1); tau(phi(p)) = tau(p-1); phi(rad(p)) = phi(p) = p-1.
Example: 2039 is prime and is solution for k = 6, and 4079 = 2*2039 + 1 is prime and is solution for n = 7; idem with the primes 32633, 913739, but p = 36549581 is prime and solution for 12 iterations, but 2*p + 1 is not prime, so it is not a solution.

			For n=1, phi(phi(7)) = 2, tau(phi(7)) = 4, phi(rad(7)) = phi(7) = 6, then 2 + 4 = 6.
For n=2, phi(phi(phi(phi(33)))) = 2, tau(phi(tau(phi(33)))) = 2, phi(rad(phi(rad(33)))) = 4, then 2 + 2 = 4.
For n=3, phi(phi(phi(phi(phi(phi(29)))))) = 1, tau(phi(tau(phi(tau(phi(29)))))) = 1, phi(rad(phi(rad(phi(rad(29)))))) = 2, then 1 + 1 = 2.

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.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. K. Caldwell, The Prime Glossary, Number of divisors
Wikipedia, Euler's totient function

Cf. A000010 (phi), A007947 (rad), A000005 (tau), A002183.

with(numtheory):for n from 1 to 100 do:indic:=0:for x from 1 to 10000 while(indic=0 ) do:x0:=x:y0:=x:z0:=x: for iter from 1 to n do:x1:=phi(phi(x0)): y1:= tau(phi(y0)): zz1:= ifactors(z0)[2] : zz2 :=mul(zz1[i][1], i=1..nops(zz1)): z1:=phi(zz2):x0:=x1:y0:=y1:z0:=z1:od :if x0 +y0=z0 then print (x):indic:=1:else fi:od:od:

rad(m) = factorback(factorint(m)[, 1]); \\ A007947
phi_phi(m,n) = {for (k=1, n, m = eulerphi(eulerphi(m));); m;}
tau_phi(m,n) = {for (k=1, n, m = numdiv(eulerphi(m));); m;}
phi_rad(m,n) = {for (k=1, n, m = eulerphi(rad(m));); m;}
a(n) =  {my(m=1); while (phi_phi(m,n)+ tau_phi(m,n) != phi_rad(m,n), m++); m;} \\ Michel Marcus, Sep 17 2020

Edited by Michel Marcus, Sep 17 2020

A333328 Irregular triangle read by rows: T(n,0) = A002182(n) and T(n,k + 1) = A000005(T(n,k)), terminating at the first number which is not highly composite, n > 2.

Original entry on oeis.org

4, 3, 6, 4, 3, 12, 6, 4, 3, 24, 8, 36, 9, 48, 10, 60, 12, 6, 4, 3, 120, 16, 180, 18, 240, 20, 360, 24, 8, 720, 30, 840, 32, 1260, 36, 9, 1680, 40, 2520, 48, 10, 5040, 60, 12, 6, 4, 3, 7560, 64, 10080, 72, 15120, 80, 20160, 84, 25200, 90, 27720, 96, 45360, 100

Offset: 3

Davis Smith, Mar 15 2020

There are two questions related to this array: First, which rows have length greater than any previous row? Second, are there any rows which terminate at a k greater than 6?

			The irregular triangle T(n,k) starts:
  n\k   0   1   2   3   4   ...
   3:   4   3
   4:   6   4   3
   5:  12   6   4   3
   6:  24   8
   7:  36   9
   8:  48  10
   9:  60  12   6   4   3
  10: 120  16
  11: 180  18
  12: 240  20
  13: 360  24   8
  ...

James Grime and Brady Haran, 5040 and other Anti-Prime Numbers, Numberphile video (2016).

Cf. A000005, A002182, A002183, A189394.

A333328_rows(n)={my(N=Map(Mat([1,1;2,2;m=4,3])),p=2,F=[]); while(#Np,mapput(N,m,p=numdiv(m)); my(M=List([m,q=p])); while(mapisdefined(N,q,&q),listput(M,q));print(#N", "Vec(M)); F=concat(F,Vec(M))); my(s=if(m>=720720,360360,m>=5040,2520,m>=840,420,m>=60,60,2)); until(numdiv(m+=s)>p,));F}

T(n,0) = A002182(n), T(n,k) = A000005(T(n,k - 1)).

A350641 Numbers k such that the product of k and all terms < k in A050376 has more divisors than the product of all terms < k in A050376 and the smallest term > k in A050376.

Original entry on oeis.org

42, 66, 72, 78, 88, 104, 110, 130, 136, 152, 156, 160, 170, 184, 190, 200, 204, 224, 228, 230, 232, 238, 240, 248, 255, 285, 345, 435, 460, 465, 483, 525, 555, 580, 600, 609, 615, 620, 651, 696, 744, 777, 783, 812, 837, 861, 868, 888, 903, 930, 984, 987, 999

Offset: 1

J. Lowell, Jan 09 2022

nonn

Multiplying a number in this sequence by all numbers in A050376 less than it will give a number less than, but with more divisors than, a number in A037992 with comparable magnitude.

			The product of 42 and all terms < 42 in A050376 has 276480 divisors. The product of all terms < 42 in A050376 and the smallest term > 42 (i.e., 43) in A050376 has only 262144 divisors. Thus, 42 is a term of this sequence.

Cf. A050376, A037992, A002183.

list(lim) = my(v=primes(primepi(lim)), t); forprime(p=2, sqrt(lim), t=p; while((t=t^2)<=lim, v=concat(v, t))); vecsort(v); \\ A050376
lista(nn) = my(vfd=list(nn), res=List()); for (n=1, nn, my(vless = select(x->(x(x>n), vfd)); if (#vmore, my(p = vecprod(vless)); if (numdiv(p*n) > numdiv(p*vmore[1]), listput(res, n));););); res; \\ Michel Marcus, Jan 10 2022

More terms from Jinyuan Wang, Jan 09 2022

cp's OEIS Frontend

A095717 "Second order" highly composite numbers: the gap between the number of divisors (d(n)) rises to a new record.

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A141320 Both n and the smallest number with n divisors are in A002182.

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A161887 A product of quotients of factorials.

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A243423 Number of divisors of A243220(n).

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A342833 Integers m such that the number of divisors whose last digit equals the last digit of m sets a new record.

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A346729 Maximum number of divisors among n-bit numbers.

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A356078 Highly composite numbers that are multiples of their number of divisors.

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A175189 Smallest integer m such that phi(phi(m))^n + tau(phi(m))^n = phi(rad(m))^n, where n is the number of iterations of phi(phi), tau(phi) and phi(rad) functions.

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