A174228 -id:A174228 - cp's OEIS frontend

A018253 Divisors of 24.

1, 2, 3, 4, 6, 8, 12, 24

Offset: 1

The divisors of 24 greater than 1 are the only positive integers n with the property m^2 == 1 (mod n) for all integer m coprime to n. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001
Numbers n for which all Dirichlet characters are real. - Benoit Cloitre, Apr 21 2002
These are the numbers n that are divisible by all numbers less than or equal to the square root of n. - Tanya Khovanova, Dec 10 2006 [For a proof, see the Tauvel paper in references. - Bernard Schott, Dec 20 2012]
Also, numbers n such that A160812(n) = 0. - Omar E. Pol, Jun 19 2009
It appears that these are the only positive integers n such that A160812(n) = 0. - Omar E. Pol, Nov 17 2009
24 is a highly composite number: A002182(6)=24. - Reinhard Zumkeller, Jun 21 2010
Chebolu points out that these are exactly the numbers for which the multiplication table of the integers mod n have 1s only on their diagonal, i.e., ab == 1 (mod n) implies a = b (mod n). - Charles R Greathouse IV, Jul 06 2011
It appears that 3, 4, 6, 8, 12, 24 (the divisors >= 3 of 24) are also the only numbers n whose proper non-divisors k are prime numbers if k = d-1 and d divides n. - Omar E. Pol, Sep 23 2011
About the last Pol's comment: I have searched to 10^7 and have found no other terms. - Robert G. Wilson v, Sep 23 2011
Sum_{i=1..8} A000005(a(i))^3 = (Sum_{i=1..8} A000005(a(i)))^2, see Kordemsky in References and Barbeau et al. in Links section. - Bruno Berselli, Dec 29 2014

			Square root of 12 = 3.46... and 1, 2 and 3 divide 12.
From the tenth comment: 1^3 + 2^3 + 2^3 + 3^3 + 4^3 + 4^3 + 6^3 + 8^3 = (1+2+2+3+4+4+6+8)^2 = 900. - _Bruno Berselli_, Dec 28 2014

Harvey Cohn, "Advanced Number Theory", Dover, chap.II, p. 38
Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations, C. Scribner's Sons (1972), Chapter XIII, Paragraph 349.
Patrick Tauvel, "Exercices d'algèbre générale et d'arithmétique", Dunod, 2004, exercice 70 page 368.

John Baez, My Favorite Number: 24, The Rankin Lectures (September 19, 2008).
Edward Barbeau and Samer Seraj, Sum of Cubes is Square of Sum, arXiv:1306.5257 [math.NT], 2013.
Paul T. Bateman and Marc E. Low, Prime numbers in arithmetic progressions with difference 24, The American Mathematical Monthly 72:2 (Feb., 1965), pp. 139-143.
Sunil K. Chebolu, What is special about the divisors of 24?, Math. Mag., 85 (2012), 366-372.
M. H. Eggar, A curious property of the integer 24, Math. Gazette 84 (2000), pp. 96-97.
J. C. Lagarias (proposer), Problem 11747, Amer. Math. Monthly, 121 (2014), 83.
Eric Weisstein's World of Mathematics, Modulo Multiplication Group
Index entries for sequences related to divisors of numbers

Cf. A174228, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858-A178864.
Cf. A000005, A158649. - Bruno Berselli, Dec 29 2014
Cf. A303704 (with respect to Astudillo's 2001 comment above).

DivisorsInt(24); # Bruno Berselli, Feb 13 2018

Divisors[24] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(24) \\ Charles R Greathouse IV, Apr 28 2011

divisors(24); # Zerinvary Lajos, Jun 13 2009

a(n) = A161710(n-1). - Reinhard Zumkeller, Jun 21 2009

A080683 23-smooth numbers: numbers whose prime divisors are all <= 23.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95

Offset: 1

Cino Hilliard, Mar 02 2003

Coincides for the first 111 terms with A174228 (divisors of 24!). - Bruno Berselli, Sep 24 2012

William A. Tedeschi, Table of n, a(n) for n = 1..10000

For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080197, A080681, A080682.

[n: n in [1..100] | PrimeDivisors(n) subset PrimesUpTo(23)]; // Bruno Berselli, Sep 24 2012

select(t -> max(numtheory:-factorset(t)) <= 23, [$1..1000]); # Robert Israel, Jan 22 2016

mx = 100; Sort@ Flatten@ Table[ 2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i, {a, 0, Log[2, mx]}, {b, 0, Log[3, mx/2^a]}, {c, 0, Log[5, mx/(2^a*3^b)]}, {d, 0, Log[7, mx/(2^a*3^b*5^c)]}, {e, 0, Log[11, mx/(2^a*3^b*5^c*7^d)]}, {f, 0, Log[13, mx/(2^a*3^b*5^c*7^d*11^e)]}, {g, 0, Log[17, mx/(2^a*3^b*5^c*7^d*11^e*13^f)]}, {h, 0, Log[19, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g)]}, {i, 0, Log[23, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h)]}] (* Robert G. Wilson v, Jan 19 2016 *)

test(n)=m=n; forprime(p=2,23, while(m%p==0,m=m/p)); return(m==1)
for(n=1,100,if(test(n),print1(n",")))

list(lim,p=23)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=23): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 72))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A080683(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,23)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 23} p/(p-1) = (2*3*5*7*11*13*17*19*23)/(1*2*4*6*10*12*16*18*22) = 676039/110592. - Amiram Eldar, Sep 22 2020

A155182 Divisors of 12!.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 99, 100, 105, 108, 110, 112, 120, 126, 128, 132, 135, 140, 144, 150, 154, 160

Offset: 1

Reinhard Zumkeller, Jan 21 2009

The sequence is finite with A027423(12) = 792 terms:

a(792) = A000142(12) = 479001600 is the last term.

			Last ten numbers: 47900160, 53222400, 59875200, 68428800, 79833600, 95800320, 119750400, 159667200, 239500800, 479001600. - _Zerinvary Lajos_, Jun 13 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..792 (full sequence)
Index entries for sequences related to divisors of numbers.
Index entries for sequences related to factorial numbers.

Subsequence of A051038.

Cf. A174228. - Reinhard Zumkeller, May 24 2010

Divisors[12!][[;;100]] (* Paolo Xausa, Jul 31 2024 *)

PARI
```
divisors(12!)
```

divisors(factorial(12)) # Zerinvary Lajos, Jun 13 2009

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A018253 Divisors of 24.

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A080683 23-smooth numbers: numbers whose prime divisors are all <= 23.

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A155182 Divisors of 12!.

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