A139770 Smallest number having at least as many divisors as n.

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

Offset

1,2

Comments

Similar to A140635, except that a(n) is allowed to have more divisors than n.

a(n) <= n for all n. Moreover, a(n) = n if and only if n belongs to A061799 (or equivalently A002182).

When n is prime, a(n) = 2. - Michel Marcus, Jun 14 2013

For numbers k such that a(k) and A140635(k) are not equal see A365263. - Michel Marcus, Aug 31 2023

Links

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

Formula

a(n) = A061799(A000005(n)).

Example

16 has 5 divisors; smallest number with at least 5 divisors is 12 with 6 divisors, thus a(16) = 12.

Mathematica

a139770[n_] := NestWhile[#+1&, 1, DivisorSigma[0, n]>DivisorSigma[0, #]&]
a139770[{m_, n_}] := Map[a139770, Range[m, n]]
a139770[{1, 89}] (* Hartmut F. W. Hoft, Jun 13 2023 *)

Prog

(PARI) a(n) = {nd = numdiv(n); for (i=1, n-1, if (numdiv(i) >= nd, return (i)); ); return (n); } \\ Michel Marcus, Jun 14 2013
(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, A002182, A061799, A140635, A365263.

Sequence in context: A122457 A319410 A337174 * A140635 A283465 A283466

Adjacent sequences: A139767 A139768 A139769 * A139771 A139772 A139773

Keyword

nonn

Author

J. Lowell, May 20 2008

Extensions

Edited and extended by Ray Chandler, May 24 2008

Status

approved