A140635 Smallest positive integer having the same number of divisors as n.

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

Offset: 1

Author

Max Alekseyev, May 19 2008

Keywords

• nonn

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

Crossrefs

Cf. A000005, A005179, A007416, A139770.

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

Programs

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 *)

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

Formula

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