A140635 -id:A140635 - cp's OEIS frontend

A365264 a(n) is the smallest positive integer k whose number of divisors is larger than that of n.

2, 4, 4, 6, 4, 12, 4, 12, 6, 12, 4, 24, 4, 12, 12, 12, 4, 24, 4, 24, 12, 12, 4, 36, 6, 12, 12, 24, 4, 36, 4, 24, 12, 12, 12, 48, 4, 12, 12, 36, 4, 36, 4, 24, 24, 12, 4, 60, 6, 24, 12, 24, 4, 36, 12, 36, 12, 12, 4, 120, 4, 12, 24, 24, 12, 36, 4, 24, 12, 36, 4, 120, 4, 12, 24, 24, 12, 36, 4, 60

Offset: 1

Hartmut F. W. Hoft, Aug 29 2023

nonn

The distinct values in this sequence together with 1 form A002182.

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

Cf. A002182, A139770, A140635.

a365264[n_] := NestWhile[#+1&, 1, DivisorSigma[0, n]>=DivisorSigma[0, #]&]
a365264[{m_, n_}] := Map[a365264, Range[m, n]]
a365264[{1, 80}]

a(n) = my(k=1, nd=numdiv(n)); while(numdiv(k) <= nd, k++); k; \\ Michel Marcus, Aug 29 2023

a(n) = Min_{ k >= 1, sigma_0(k) > sigma_0(n) }, n>=1.

A002182(n+1) = a(a(...a(1))...) with n >= 0 iterations.

A241813 Numbers disqualified from being in A019505 for not being the smallest number with their respective number of divisors.

Original entry on oeis.org

8, 96, 480, 1440, 40320, 443520, 1330560, 34594560, 86486400, 588107520, 1470268800, 11174042880, 55870214400, 195545750400, 1285014931200, 17990209036800, 53970627110400, 1565148186201600, 194078375088998400, 7180899878292940800, 35904499391464704000, 294416895010010572800

Offset: 1

J. Lowell, Apr 29 2014

It appears that when 2*A019505(n) is a member of this sequence then the exponent in at least one primary of the factorization of A019505(n+1) is smaller than in the corresponding primary of A019505(n) or A019505(n+1) contains an additional prime factor. The smallest example in this sequence where two primaries have smaller exponents and an additional prime factor is added is a(14) = 2*A019505(43) = 2 * 97772875200 = 195545750400. The sequence of exponents of its primaries is (7, 3, 2, 2, 1, 1, 1, 1 ) while A019505(44) = 160626866400 has exponent sequence (5, 3, 2, 1, 1, 1, 1, 1, 1 ). - Hartmut F. W. Hoft, Feb 22 2023

			8 qualifies because 8 = 4*2 and 4 is in A019505, but 8 can't be term after 4 in A019505 because smallest number with 4 divisors is 6.

Cf. A019505, A140635.

dataA019505 = Map[Last, Import[URL["https://oeis.org/A019505/b019505.txt"], "Data"]]
dataA241813 = Take[Map[First, Select[Map[{2#[[1]], 2#[[1]]==#[[2]]}&, Transpose[{Most[dataA019505], Rest[dataA019505]}]], !#[[2]]&]], 22] (* Hartmut F. W. Hoft, Feb 22 2023 *)

More terms from Hartmut F. W. Hoft, Feb 22 2023

cp's OEIS Frontend

A365264 a(n) is the smallest positive integer k whose number of divisors is larger than that of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula