A064145 -id:A064145 - cp's OEIS frontend

A104361 Number of divisors of A104357(n) = A104350(n) - 1.

1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 4, 4, 2, 4, 8, 8, 2, 8, 8, 8, 2, 4, 16, 16, 4, 4, 32, 16, 4, 8, 4, 8, 2, 8, 16, 8, 32, 8, 16, 16, 32, 2, 8, 16, 4, 4, 4, 8, 2, 4, 8, 4, 8, 2, 16, 2, 8, 16, 64, 16, 16, 4, 64, 2, 32, 16, 16, 2, 4, 32, 128, 16, 32, 8, 16, 16, 32, 16, 32, 4, 16, 16, 32, 8, 8, 8, 4, 32, 4, 32, 16, 128, 4, 32, 8, 4

Offset: 2

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 2..145
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000005, A064145, A104350, A104357, A104358, A104359, A104360, A104362, A104363, A104364, A104369.

a[n_] := DivisorSigma[0, -1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]]; Array[a, 50, 2] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A000005(A104357(n)).

a(51)-a(74) from Amiram Eldar, Feb 12 2020

Terms a(75) onward from Max Alekseyev, Oct 03 2022

A064144 a(n) is the number of divisors of n!+1.

Original entry on oeis.org

2, 2, 2, 3, 3, 4, 3, 4, 8, 4, 2, 6, 4, 4, 8, 32, 8, 64, 4, 4, 8, 8, 12, 4, 4, 4, 2, 4, 8, 32, 16, 16, 32, 4, 32, 64, 2, 4, 16, 128, 2, 8, 16, 8, 8, 8, 16, 4, 32, 32, 64, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 8, 32, 8

Offset: 1

Vladeta Jovovic, Sep 11 2001

nonn

Max Alekseyev, Table of n, a(n) for n = 1..139
William Gerst, A conjecture on the prime factorization of n!+1, arXiv:1809.07360 [math.GM], 2018.
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002583, A027423, A002981, A002982, A064145, A078778, A181764.

Do[ Print[ DivisorSigma[0, n! + 1]], {n, 1, 40} ]

a(n) = numdiv(n! + 1); \\ Harry J. Smith, Sep 09 2009

from math import factorial
from sympy import divisor_count
def A064144(n): return divisor_count(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

a(n) = tau(n!+1).

More terms from Robert G. Wilson v, Oct 04 2001
a(42)-a(64) from Harry J. Smith, Sep 09 2009
Edited by Jon E. Schoenfield, Jun 21 2018

A366757 a(n) is the sum of the divisors of n!-1.

Original entry on oeis.org

1, 6, 24, 144, 720, 5040, 42096, 399000, 3753960, 47500992, 479001600, 6230615736, 87178291200, 1457696910960, 20929670124480, 379536693283440, 6510917252872320, 121831439598033840, 2432921507427445440, 53921727651043042560, 1134312679767378217920

Offset: 2

Sean A. Irvine, Oct 20 2023

nonn

			a(5) = 144 because the divisors of 5!-1 are {1, 7, 17, 119}.

Cf. A033312, A000203, A064145, A062569, A366758.

a:=n->numtheory[sigma](n!-1):
seq(a(n), n=2..30);

DivisorSigma[1,Range[2,25]!-1] (* Paolo Xausa, Oct 21 2023 *)

from math import factorial
from sympy import divisor_sigma
def A366757(n): return divisor_sigma(factorial(n)-1) # Chai Wah Wu, Oct 20 2023

a(n) = sigma(n!-1) = A000203(A033312(n)).

A366809 The sum of the divisors of prime(n)#-1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

1, 6, 30, 240, 2310, 30030, 518940, 9943560, 230876448, 6551588160, 200561595684, 7471933410000, 304250263527210, 13082853940673340, 618109122639794688, 32589631537463089128, 1922760350251477679196, 117386696543681561301312, 7906535060701218163040640

Offset: 1

Sean A. Irvine, Oct 23 2023

nonn

			a(4)=240 because the divisors of 7#-1 = 209 are {1, 11, 19, 209}.

Cf. A057588, A000230, A054989, A366809, A064145.

seq(numtheory[sigma](mul(ithprime(k), k=1..n)-1), n=1..30);

a(n) = sigma(prime(n)#-1) = A000230(A057588(n)).

A366811 The number of divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 4, 8, 4, 4, 8, 2, 8, 8, 4, 8, 16, 16, 4, 4, 16, 4, 8, 4, 16, 8, 4, 16, 16, 8, 8, 32, 8, 64, 4, 8, 4, 32, 16, 16, 4, 64, 8, 16, 8, 32, 64, 128, 4, 64, 8, 32, 8, 16, 4, 64, 32, 16, 32, 8, 32, 32, 32, 8, 8, 32, 32, 64, 8, 16, 16, 128, 32, 8, 16

Offset: 0

Sean A. Irvine, Oct 23 2023

nonn

			a(6) = 4 because the divisors of 13#+1 = 30031 are {1, 59, 509, 30031}.

Amiram Eldar, Table of n, a(n) for n = 0..98

Cf. A006862, A000005, A054988, A366808, A366812, A064145.

seq(numtheory[tau](mul(ithprime(k), k=1..n)+1), n=0..30);

Map[DivisorSigma[0, #] &, 1 + FoldList[Times, 1, Prime@ Range@ 19] ] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sigma0(prime(n)#+1) = A000005(A006862(n)).

A331547 Numbers k such that k and k! - 1 have the same number of divisors.

Original entry on oeis.org

3, 7, 8, 10, 26, 27, 34, 85, 93, 104, 143, 152

Offset: 1

Matthew Niemiro, Jan 20 2020

The sequence also includes: 143, 152, 186, 230, 379, 381, 543, 573, 602. - Daniel Suteu, Jan 21 2020
The sequence also includes 2881. Even though the complete factorization of 136!-1 is not known, 136 is not a term, since 136!-1 is known to be the product of 2 distinct primes and a composite number, so it has at least 4 prime factors and 3 distinct prime factors, thus the number of divisors >= 12, whereas 136 has 8 divisors. - Chai Wah Wu, Feb 26 2020
Similar reasoning (considering only small prime factors of k! - 1) shows that the next terms (> a(12) = 152) can only be within the set {154, 160, 162, 164, 176, 180, 182, 186, 187, 188, 192, 195, 196, 198, 204, ...}. - M. F. Hasler, Feb 26 2020

factordb.com, Status of 154!-1.

Supersequence of A103317.

Cf. A000005, A002982, A064145.

Select[Range[50], DivisorSigma[0, #] - DivisorSigma[0, Factorial[#] - 1] == 0 &]

isok(k) = k>1 && numdiv(k)==numdiv(k!-1); \\ Jinyuan Wang, Jan 20 2020

{is(n)=my(f); n>2&& numdiv(n)>=numdiv(f=factor(n!-1,0))&& if( ispseudoprime(vecmax(f[,1])), numdiv(n)==numdiv(f), numdiv(n)<2*numdiv(f), 0, numdiv(n)==numdiv(n!-1))} \\ Avoids complete factorization if possible. - M. F. Hasler, Feb 26 2020

A331547 = { n > 1 | A000005(n) = A064145(n) }. - M. F. Hasler, Feb 26 2020

a(8)-a(9) from Jinyuan Wang, Jan 20 2020
a(10) from Amiram Eldar, Jan 20 2020
a(11)-a(12) from Chai Wah Wu, Feb 26 2020

A366808 The number of divisors of prime(n)#-1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 4, 8, 8, 4, 4, 16, 2, 4, 8, 8, 4, 8, 8, 4, 4, 16, 8, 2, 4, 4, 16, 16, 16, 8, 8, 8, 8, 8, 8, 16, 16, 16, 32, 8, 32, 16, 32, 16, 16, 16, 16, 4, 8, 8, 4, 16, 8, 16, 4, 16, 16, 128, 16, 8, 8, 16, 16, 8, 8, 2, 8, 2, 16, 8, 32, 32, 16, 16, 64, 32

Offset: 1

Sean A. Irvine, Oct 23 2023

nonn

			a(4)=4 because the divisors of 7#-1 = 209 are {1, 11, 19, 209}.

Cf. A057588, A000005, A054989, A366809, A064145.

seq(numtheory[tau](mul(ithprime(k), k=1..n)-1), n=1..30);

Map[DivisorSigma[0, #] &, -1 + FoldList[Times, Prime@ Range@ 30] ] (* Michael De Vlieger, Oct 25 2023 *)

a(n) = sigma0(prime(n)#-1) = A000005(A057588(n)).

cp's OEIS Frontend

A104361 Number of divisors of A104357(n) = A104350(n) - 1.

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A064144 a(n) is the number of divisors of n!+1.

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A366757 a(n) is the sum of the divisors of n!-1.

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A366809 The sum of the divisors of prime(n)#-1 where p# is the product of all the primes from 2 to p inclusive.

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A366811 The number of divisors of prime(n)#+1 where p# is the product of all the primes from 2 to p inclusive.

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A331547 Numbers k such that k and k! - 1 have the same number of divisors.

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A366808 The number of divisors of prime(n)#-1 where p# is the product of all the primes from 2 to p inclusive.

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