A061300 -id:A061300

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A357582

a(n) = A061300(n+1)/A061300(n).

+20
1

1, 2, 6, 30, 154, 1105, 4788, 20677, 216931, 858925, 7105392, 5546059, 2018025900, 1480452337, 3238556831, 107972737, 18425956230000, 4683032671, 14053747110612300, 160436746661, 33809725025123, 15260431896321667, 1583855315457687090000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

It is not known if the ratios A061300(n+1)/A061300(n) are always integer, but so far (for the listed terms) they are. - Max Alekseyev, Sep 05 2023

LINKS

Max Alekseyev, Table of n, a(n) for n = 0..29

EXAMPLE

a(5) = 1105 as A061300(5+1) / A061300(5) = 61261200 / 55440 = 1105.

MATHEMATICA

A061300[n_Integer?NonNegative] := A061300[n] = Module[{fact = n!, num = 1}, Monitor[While[Length@Divisors@num != fact, num++]; num, {n, num}]]; a[n_] := A061300[n + 1]/A061300[n]; Table[a[n], {n, 0, 4}] (* Robert P. P. McKone, Sep 07 2023 *)

CROSSREFS

Cf. A061300.

KEYWORD

nonn,hard

AUTHOR

J. Lowell, Oct 04 2022

EXTENSIONS

a(11)-a(21) from David A. Corneth, Oct 05 2022

a(22)-a(29) from Max Alekseyev, Sep 05 2023

STATUS

approved

A061307

Duplicate of A061300.

+20
0

1, 1, 2, 12, 360, 55440, 61261200, 293318625600, 6064949221531200

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..8.

KEYWORD

dead

STATUS

approved

A140635

Smallest positive integer having the same number of divisors as n.

+10
13

1, 2, 2, 4, 2, 6, 2, 6, 4, 6, 2, 12, 2, 6, 6, 16, 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, 64, 6, 24, 2, 12, 6, 24, 2, 60, 2, 6, 12, 12, 6, 24, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) <= n for all n. Moreover, a(n) = n if and only if n belongs to A005179 or A007416.

LINKS

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

FORMULA

a(n) = A005179(A000005(n)).

MATHEMATICA

a140635[n_] := NestWhile[#+1&, 1, DivisorSigma[0, n]!=DivisorSigma[0, #]&]

a140635[{m_, n_}] := Map[a140635, Range[m, n]]

a140635[{1, 89}] (* Hartmut F. W. Hoft, Jun 13 2023 *)

PROG

(PARI) A140635(n) = { my(nd = numdiv(n)); for (i=1, n, if (numdiv(i) == nd, return (i))); }; \\ After A139770, Antti Karttunen, May 27 2017

(Python)

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

CROSSREFS

Cf. A000005, A005179, A007416, A139770.

Cf. A019505, A138113, A061300 (sequences that can be defined in terms of this sequence).

KEYWORD

nonn

AUTHOR

Max Alekseyev, May 19 2008

STATUS

approved

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