A068499 -id:A068499 - cp's OEIS frontend

A027423 Number of divisors of n!.

1, 1, 2, 4, 8, 16, 30, 60, 96, 160, 270, 540, 792, 1584, 2592, 4032, 5376, 10752, 14688, 29376, 41040, 60800, 96000, 192000, 242880, 340032, 532224, 677376, 917280, 1834560, 2332800, 4665600, 5529600, 7864320, 12165120, 16422912

Offset: 0

Glen Burch (gburch(AT)erols.com), Leroy Quet

It appears that a(n+1)=2*a(n) if n is in A068499. - Benoit Cloitre, Sep 07 2002
Because a(0) = 1 and for all n > 0, 2*a(n) >= a(n+1), the sequence is a complete sequence. - Frank M Jackson, Aug 09 2013
Luca and Young prove that a(n) divides n! for n >= 6. - Michel Marcus, Nov 02 2017

			a(4) = 8 because 4!=24 has precisely eight distinct divisors: 1, 2, 3, 4, 6, 8, 12, 24.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Daniel Berend and J. E. Harmse, Gaps between consecutive divisors of factorials, Ann. Inst. Fourier, 43 (3) (1993), 569-583.
Paul Erdős, S. W. Graham, Alexsandr Ivić, and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, ed. by B. C. Berndt, H. G. Diamond, A. J. Hildebrand, Birkhauser 1996, pp. 337-355.
Florian Luca and Paul Thomas Young, On the number of divisors of n! and of the Fibonacci numbers, Glasnik Matematicki, Vol. 47, No. 2 (2012), 285-293. DOI: 10.3336/gm.47.2.05.
John D. Mahony, The next number in the sequence, Math. Gaz. (2024) Vol. 108, Issue 573, 399-406.
Wikipedia, Complete sequence.
Index entries for sequences related to factorial numbers
Index entries for sequences related to divisors of numbers

Cf. A000005, A000142, A062569, A131688, A161466 (divisors of 10!).

a027423 n = f 1 $ map (\p -> iterate (* p) p) a000040_list where
   f y ((pps@(p:_)):ppss)
     | p <= n = f (y * (sum (map (div n) $ takeWhile (<= n) pps) + 1)) ppss
     | otherwise = y
-- Reinhard Zumkeller, Feb 27 2013
(Python 3.8+)
from math import prod
from collections import Counter
from sympy import factorint
def A027423(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()) # Chai Wah Wu, Jun 25 2022

Maple
```
A027423 := n -> numtheory[tau](n!);
```

Table[ DivisorSigma[0, n! ], {n, 0, 35}]

for(k=0,50,print1(numdiv(k!),", ")) \\ Jaume Oliver Lafont, Mar 09 2009

a(n)=my(s=1,t,tt);forprime(p=2,n,t=tt=n\p; while(tt, t+=tt\=p); s*=t+1); s \\ Charles R Greathouse IV, Feb 08 2013

a(n) <= a(n+1) <= 2*a(n) - Benoit Cloitre, Sep 07 2002
From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 28 2009: (Start)
Assume, p1,p2...pm are the prime numbers less than or equal to n.
Then, a(n) = Product_{i=1..m} (bi+1), where bk = Sum_{i=1..m} floor(n/pk^i).
For example, if n=5, p1=2,p2=3,p3=5;
b1=floor(5/2)+floor(5/2^2)+floor(5/2^3)+...=2+1+0+..=3 similarly, b2=b3=1;
Thus a(5)=(3+1)(1+1)(1+1)=16. (End)
a(n) = A000005(A000142(n)). - Michel Marcus, Sep 13 2014
a(n) ~ exp(c * n/log(n) + O(n/log(n)^2)), where c = A131688 (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

A337536 Numbers k for which there are only 2 bases b (2 and k+1) where the digits of k contain the digit b-1.

Original entry on oeis.org

2, 3, 4, 10, 36, 40, 82, 256

Offset: 1

Michel Marcus, Aug 31 2020

These could be called "nine-free numbers".
From David A. Corneth, Aug 31 2020: (Start)
This sequence has density 0. Conjecture: this sequence is finite and full. a(9) > 10^100 if it exists.
Suppose we want to see if 22792 = 1011021011_3 is a term. Since it has a digit of 2 in base 3, we can see that it is not. The next number that does not have the digit 2 in base 3 is 1011100000_3 = 22842, so we can proceed from there. In a similar way we can skip numbers based on bases b > 3. (End)
All terms of this sequence increased by 1 (except a(2)=3) are prime. - François Marques, Aug 31 2020
From Devansh Singh, Sep 19 2020: (Start)
If n is one less than an odd prime and we are interested in bases 3 <= b <= n-1 such that n in base b contains the digit b-1, then divisor of b (except 1) -1 cannot be the last digit since divisor of b divides n+1, which is not possible as n+1 is an odd prime.
If the last digit is 1, then b is odd as 1 = 2-1 and 2 cannot divide b as n+1 is an odd prime.
If the last digit is 0, then b-1 is the last digit of n-1 in base b.
b <= n/2 for even n,b <= (n+1)/2 for odd n.
This sequence is equivalent to the existence of only one prime generating polynomial = F(x) (having positive integer coefficients >=0 and <=b-1 for F(b)) such that F(2) = p.
There is no other prime generating polynomial = G(x) (having positive integer coefficients >=0 and <= b-1 for G(b)) that generates p for 2 < x = b <= (p-1)/2.
(End)

			2 is a term because 2 = 10_2 = 2_3, so both have the digit b-1, and there are no other bases where this happens.
4 is a term because 4 = 100_2 = 4_5, so both have the digit b-1, and there are no other bases where this happens.

David A. Corneth, PARI program

Cf. A005836, A007095, A023717, A020654, A020657, A037465, A037474, A037477, A337496, A337535.

Subsequence of A005836 U {2} and of A068499.

isok(n, b) = vecmax(digits(n, b)) == b-1;
b(n) = if (n==1, return (1)); my(b=3); while(!isok(n, b), b++); b; \\ A337535
is(n) = b(n) == n+1;

\\ See Corneth link \\ David A. Corneth, Aug 31 2020

A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 5, 2, 3, 4, 4, 2, 5, 2, 5, 4, 3, 2, 7, 3, 3, 4, 5, 2, 7, 2, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 7, 2, 5, 6, 3, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 3, 2, 11, 2, 3, 6, 6, 4, 7, 2, 5, 4, 7, 2, 11, 2, 3, 6, 5, 4, 7, 2, 9, 5, 3, 2, 11, 4, 3, 4, 7, 2, 11, 4, 5, 4, 3, 4, 11, 2, 5, 6, 8, 2, 7, 2

Offset: 1

Amarnath Murthy, Oct 25 2002

nonn

a(A001605(n)) = 2; a(A105802(n)) = n.

It is well known that if k is a divisor of n then F(k) divides F(n). Hence if n has d divisors, one expects that a(n)=d. However because F(1)=F(2)=1, there is one fewer Fibonacci divisor when n is even. So for even n, a(n)=d-1. - T. D. Noe, Jan 18 2006

			n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.

with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);

Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)

{a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */

{a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* Michael Somos, Sep 03 2006 */

a(n) = A023645(n) + 1. - T. D. Noe, Jan 18 2006
a(n) = tau(n) - [n is even] = A000005(n) - A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture of Ralf Stephan, Sep 28 2004
The number of divisors of n excluding 2.
a(2n) = A066660(n). a(2n-1) = A099774(n). - Michael Somos, Sep 03 2006
a(3*2^(Prime(n-1)-1)) = 2n + 1 for n > 3. a(3*2^A068499[n]) = 2n + 1, where A068499(n) = {1,2,3,4,6,10,12,16,18,...}. - Alexander Adamchuk, Sep 15 2006

Corrected and extended by Sascha Kurz, Jan 26 2003

Edited by N. J. A. Sloane, Sep 14 2006. Some of the comments and formulas may need to be adjusted to reflect the new offset.

A105802 Smallest m such that the m-th Fibonacci number has exactly n divisors that are also Fibonacci numbers.

Original entry on oeis.org

1, 3, 6, 15, 12, 45, 24, 36, 48, 405, 60, 315, 192, 144, 120, 945, 180, 1575, 240, 576, 3072, 295245, 360, 1296, 12288, 900, 960, 25515, 720, 14175, 840, 9216, 196608, 5184, 1260, 17325, 786432, 36864, 1680, 31185, 2880, 127575, 15360, 3600, 99225

Offset: 1

Reinhard Zumkeller, Apr 20 2005

nonn

A076985(n) = A000045(a(n)); A076984(a(n)) = n.

			n=6: a(6) = 45, A076985(6) = A000045(45) = 1134903170,
A076984(45) = #{1,2,5,34,109441,1134903170} = #{fib(1),fib(2),fib(5),fib(9),fib(21),fib(45)} = 6.

Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A068499.

t=Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 1000000}]; lst={}; n=1; While[pos=Flatten[Position[t, n, 1, 1]]; Length[pos]>0, AppendTo[lst, pos[[1]]]; n++ ]; lst (Noe)

Conjecture: a(2k+1) = 3*2^(Prime[k-1]-1) for k>3. It appears that a(2k+1) = 3*2^k for k = {1,2,3,4,6,10,12,16,18,...} = A068499[n] Numbers n such that n! reduced modulo (n+1) is not zero. - Alexander Adamchuk, Sep 15 2006

More terms from T. D. Noe, Jan 18 2006

A368548 Number of palindromic partitions of n.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 4, 6, 2, 10, 2, 8, 10, 10, 2, 18, 2, 20, 16, 12, 2, 36, 12, 14, 24, 36, 2, 60, 2, 38, 34, 18, 46, 104, 2, 20, 46, 108, 2, 122, 2, 94, 148, 24, 2, 212, 58, 116, 76, 140, 2, 232, 164, 270, 94, 30, 2, 588, 2, 32, 372, 274, 280, 420, 2, 276

Offset: 1

Michel Marcus, Feb 06 2024

nonn

For n>1, a(n) >= 2 as [n] and [1,1,..1] are both palindromic partitions. - Chai Wah Wu, Feb 07 2024

			For n=5, the palindromic partitions (as defined in Hemmer and Westrem) are [5], [2, 3], [1, 2, 2], [1, 1, 1, 1, 1]. - _Chai Wah Wu_, Feb 07 2024

Chai Wah Wu, Table of n, a(n) for n = 1..10000
David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Table 3.1 p. 5.
Chai Wah Wu, Proofs of formulas for A368548
Chai Wah Wu, Proofs of formulas for A368548 and A375783

Cf. A068499 (a(n)=2, n>1).

from itertools import count
from math import comb
from collections import Counter
from sympy.utilities.iterables import partitions
from sympy import isprime
def A368548(n):
    if n == 3 or (n>1 and isprime(n+1)): return 2
    c = 0
    for p in partitions(n):
        s, a = '', 1
        for d in sorted(Counter(p).elements()):
            s += '1'*(d-a)+'0'
            a = d
        if s[:-1] == s[-2::-1]:
            c += 1
    return c # Chai Wah Wu, Feb 06 2024

from math import comb
from sympy import divisors
def A368548(n): # faster program using formula
    x = sum(comb(d-2+((n+1)//d>>1),d-1) for d in divisors(n+1>>1,generator=True)) if n&1 else 0
    y = sum(comb((d-5>>1)+(n+1)//d,d-3>>1) for d in divisors((n+1)>>(~(n+1)&n).bit_length(),generator=True) if d>=3)<<1
    return x+y # Chai Wah Wu, Feb 08 2024

From Chai Wah Wu, Feb 08 2024: (Start)
Let x = 0 if n is even and x = Sum_{d|(n+1)/2} binomial(d-2+(n+1)/2d,d-1) if n is odd.
Let y = 2*Sum_{d|n+1, d>=3, and d is odd} binomial((d-5)/2+(n+1)/d,(d-3)/2).
Then a(n) = x+y.
a(n) = 2 if n>1 and n+1 is prime.
a(n) = (n+3)/2 if n>3 is odd and (n+1)/2 is prime.
a(2^n-1) = Sum_{i=0..n-1} binomial(2^i+2^(n-i-1)-2,2^i-1).
(End)

a(41)-a(67) from Chai Wah Wu, Feb 06 2024

A166460 Numbers k such that k + (-1)^k is not prime.

Original entry on oeis.org

0, 1, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94

Offset: 1

Juri-Stepan Gerasimov, Oct 14 2009

This is the complement of A068499 (except that both include 1 as a term).
From Don Reble, Aug 31 2021: (Start)
Proof for all k except 0, 1, 3 with cases
(i)   If k is odd and >=5, then k+1 = 2*x, 2 < x < k, k! = k*...*x*...*2*1
        A068499: k+1 divides k!            : absent
        A166460: k-1 is even and composite : present
(ii)  If k is even and k+1 is prime,
        A068499: k+1 does not divide k!    : present
        A166460: k+1 is prime              : absent
(iii) If k is even and k+1 = p^2 is the square of a (odd) prime, then k+1 >= 3p, k > 2p.
        A068499: k! = k*...*2p*...*p*...*1;
                 k+1 divides k!            : absent
        A166460: k+1 is composite          : present
(iv)  If k is even and k+1 is composite but not the square of a prime, then there are two distinct factors x*y = k+1:
                 3 <= x < y = (k+1)/x < k.
        A068499: k! = k*...*y*...*x*...*1:
                 k+1 divides k!            : absent
        A166460: k+1 is composite          : present
(End)

			0 + (-1)^0 = 1 is not prime, which adds 0 to the sequence.
5 + (-1)^5 = 4 is not prime, which adds 5 to the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002808, A072668, A141468, A060462, A118742, A068499.

Select[Range[0, 94], ! PrimeQ[# + (-1)^#] &] (* Michael De Vlieger, Sep 08 2021 *)

from sympy import composite
def A166460(n): return composite(n-1)-1 if n>2 else n-1 # Chai Wah Wu, Aug 27 2024

a(n) = A002808(n-1)-1 for n>2. - Chai Wah Wu, Aug 27 2024

0 added by R. J. Mathar, Oct 21 2009

A071637 Largest exponent k >=0 such that (n+1)^k divides n!.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 4, 0, 1, 2, 2, 0, 3, 0, 3, 2, 1, 0, 6, 2, 1, 3, 3, 0, 6, 0, 5, 2, 1, 4, 7, 0, 1, 2, 8, 0, 5, 0, 3, 9, 1, 0, 10, 3, 5, 2, 3, 0, 7, 4, 8, 2, 1, 0, 13, 0, 1, 9, 9, 4, 5, 0, 3, 2, 10, 0, 16, 0, 1, 8, 3, 6, 5, 0, 18, 9, 1, 0, 12, 4, 1, 2, 7, 0, 20, 6, 3, 2, 1, 4, 17, 0, 7, 8, 11

Offset: 1

Benoit Cloitre, Jun 25 2002

a(A068499(n)) = 0.

			12^4 divides 11! (11!/12^4=1925) but 12^5 doesn't, hence a(11)=4.

A011776(n+1) - 1.

Table[IntegerExponent[n!,n+1],{n,500}] (* Vladimir Joseph Stephan Orlovsky, Dec 26 2010 *)

for(n=1,150,s=0; while(n!%(n+1)^s==0,s++); print1(s-1,","))

cp's OEIS Frontend

A027423 Number of divisors of n!.

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A337536 Numbers k for which there are only 2 bases b (2 and k+1) where the digits of k contain the digit b-1.

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A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

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A105802 Smallest m such that the m-th Fibonacci number has exactly n divisors that are also Fibonacci numbers.

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A368548 Number of palindromic partitions of n.

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A166460 Numbers k such that k + (-1)^k is not prime.

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