A112275 -id:A112275 - cp's OEIS frontend

A079427 Least m > n having the same number of divisors as n, a(1) = 1.

1, 3, 5, 9, 7, 8, 11, 10, 25, 14, 13, 18, 17, 15, 21, 81, 19, 20, 23, 28, 22, 26, 29, 30, 49, 27, 33, 32, 31, 40, 37, 44, 34, 35, 38, 100, 41, 39, 46, 42, 43, 54, 47, 45, 50, 51, 53, 80, 121, 52, 55, 63, 59, 56, 57, 66, 58, 62, 61, 72, 67, 65, 68, 729, 69, 70, 71, 75, 74, 78, 73

Offset: 1

Reinhard Zumkeller, Jan 08 2003

nonn

tau(a(n)) = tau(n) and tau(i) <> tau(n), n < i < a(n) (tau = A000005);

			Sets of divisors for n=10,11,12,13 and 14: D(10)={1,2,5,10}, D(11)={1,11}, D(12)={1,2,3,4,6,12}, D(13)={1,13}, D(14)={1,2,7,14}: therefore a(10)=14 (#D(10)=#D(14)).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A112275, A112276.

Cf. A171937.

a[1] = 1; a[n_] := Module[{m = n+1, d=DivisorSigma[0, n]}, While[DivisorSigma[0, m] != d, m++]; m]; Array[a, 100] (* Amiram Eldar, Feb 03 2020 *)

a(n) = if (n==1, 1, my(m=n+1, nd=numdiv(n)); while(numdiv(m) != nd, m++); m); \\ Michel Marcus, Sep 14 2021

from sympy import divisors
def a(n):
    if n == 1: return 1
    divisorsn, m = len(divisors(n)), n + 1
    while len(divisors(m)) != divisorsn: m += 1
    return m
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Sep 14 2021

a(A000040(k)) = A079428(A000040(k)) = A000040(k+1), as A000005(p)=2 for primes p.

a(n) = A171937(n) + n. - Ridouane Oudra, Sep 14 2021

A112276 Smallest number greater than n having no more divisors than n; a(1) = 1.

Original entry on oeis.org

1, 3, 5, 5, 7, 7, 11, 9, 11, 11, 13, 13, 17, 15, 17, 17, 19, 19, 23, 21, 22, 23, 29, 25, 29, 27, 29, 29, 31, 31, 37, 33, 34, 35, 37, 37, 41, 39, 41, 41, 43, 43, 47, 45, 46, 47, 53, 49, 53, 51, 53, 53, 59, 55, 57, 57, 58, 59, 61, 61, 67, 65, 65, 65, 67, 67, 71, 69, 71, 71, 73, 73

Offset: 1

Reinhard Zumkeller, Sep 01 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

A065091 (the odd primes) is a subsequence.
See A112277 for numbers m such that a(m) is composite.
Cf. A000005, A065091.
Cf. A079427, A112275.

a[1] = 1; a[n_] := Module[{m = n+1, d=DivisorSigma[0, n]}, While[DivisorSigma[0, m] > d, m++]; m]; Array[a, 100] (* Amiram Eldar, Feb 03 2020 *)

A000005(a(n)) <= A000005(n) and A000005(k) > A000005(n) for n < k < a(n).

A135974 a(n) = the smallest integer m > n such that d(m) > d(n), where d(n) = number of divisors of n.

Original entry on oeis.org

2, 4, 4, 6, 6, 12, 8, 12, 10, 12, 12, 24, 14, 16, 16, 18, 18, 24, 20, 24, 24, 24, 24, 36, 26, 28, 28, 30, 30, 36, 32, 36, 36, 36, 36, 48, 38, 40, 40, 48, 42, 48, 44, 48, 48, 48, 48, 60, 50, 54, 52, 54, 54, 60, 56, 60, 60, 60, 60, 120, 62, 63, 64, 66, 66, 72, 68, 70, 70

Offset: 1

Leroy Quet, Mar 02 2008

nonn

			a(6)=12 because 6 has 4 divisors and the smallest integer > 6 which has more than 4 divisors is 12.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A112275.

with(numtheory): a:=proc (n) local m: for m from n+1 while tau(m) <= tau(n) do end do: m end proc: seq(a(n),n=1..60); # Emeric Deutsch, Mar 21 2008

a = {}; For[n = 1, n < 70, n++, i = n + 1; While[ ! DivisorSigma[0, i] > DivisorSigma[0, n], i++ ]; AppendTo[a, i]]; a (* Stefan Steinerberger, Mar 16 2008 *)
simd[n_]:=Module[{m=n+1,d=DivisorSigma[0,n]},While[DivisorSigma[0,m]<=d,m++];m]; Array[simd,70] (* Harvey P. Dale, Oct 03 2021 *)

More terms from Stefan Steinerberger, Mar 16 2008

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A079427 Least m > n having the same number of divisors as n, a(1) = 1.

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A112276 Smallest number greater than n having no more divisors than n; a(1) = 1.

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A135974 a(n) = the smallest integer m > n such that d(m) > d(n), where d(n) = number of divisors of n.

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