A171937 -id:A171937 - 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

A377319 a(n) is the smallest positive integer k such that n + k and n - k have the same number of divisors.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 3, 1, 6, 3, 2, 3, 6, 1, 1, 3, 2, 9, 5, 2, 6, 3, 3, 6, 12, 1, 4, 6, 4, 1, 5, 2, 2, 6, 2, 3, 1, 1, 8, 3, 2, 11, 3, 4, 7, 3, 1, 6, 2, 3, 1, 1, 4, 7, 9, 1, 4, 7, 4, 3, 6, 5, 2, 2, 2, 3, 6, 1, 4, 4, 4, 3, 6, 4, 9, 6, 2, 5, 5, 2, 8, 1, 3, 3, 2, 3

Offset: 4

Felix Huber, Nov 17 2024

nonn

If the strong Goldbach conjecture is true, that every even number >= 8 is the sum of two distinct primes, then a positive integer k <= A082467(n) exists for n >= 4.

			a(8) = 2 because 10 and 6 have both four divisors. 9 and 7 have a different number of divisors.

Felix Huber, Table of n, a(n) for n = 4..10000

Cf. A000005, A067888 , A082467, A171937, A377320, A377321.

A377319:=proc(n)
   local k;
   for k to n-1 do
      if NumberTheory:-tau(n+k)=NumberTheory:-tau(n-k) then
         return k
      fi
   od;
end proc;
seq(A377319(n),n=4..90);

A377319[n_] := Module[{k = 0}, While[DivisorSigma[0, ++k + n] != DivisorSigma[0, n - k]]; k];
Array[A377319, 100, 4] (* Paolo Xausa, Dec 03 2024 *)

a(n) = my(k=1); while (numdiv(n+k) != numdiv(n-k), k++); k; \\ Michel Marcus, Nov 17 2024

1 <= a(n) <= A082467(n).

A171936 Backwards van Eck transform of A000005.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 2, 5, 2, 4, 0, 2, 4, 1, 0, 4, 6, 2, 2, 6, 1, 4, 0, 16, 4, 1, 8, 6, 6, 2, 4, 6, 1, 1, 0, 6, 3, 1, 10, 4, 2, 2, 12, 1, 7, 4, 0, 24, 5, 5, 2, 6, 12, 4, 2, 2, 1, 6, 0, 2, 4, 11, 0, 3, 10, 6, 5, 4, 4, 4, 12, 2, 5, 7, 1, 3, 8, 6, 32, 65, 5, 4, 12, 3, 1, 1, 10, 6, 6, 4, 16, 2, 1, 1, 6, 8, 6, 1

Offset: 1

N. J. A. Sloane, Oct 24 2010

nonn

N. J. A. Sloane, Transforms

Cf. A171937.

A275479 Least positive k such that d(n) divides d(n+k) (d = A000005).

Original entry on oeis.org

1, 1, 2, 5, 1, 2, 1, 2, 3, 4, 1, 6, 1, 1, 6, 32, 1, 2, 1, 8, 1, 2, 1, 6, 3, 1, 3, 4, 1, 10, 1, 12, 1, 1, 3, 64, 1, 1, 1, 2, 1, 12, 1, 1, 5, 5, 1, 32, 1, 2, 3, 8, 1, 2, 1, 10, 1, 2, 1, 12, 1, 3, 5, 128, 1, 4, 1, 4, 1, 8, 1, 12, 1, 3, 1, 8, 1, 10, 1, 32, 31, 2, 1, 6, 1, 1, 1, 14, 1, 6, 2, 4, 1, 1, 1, 12, 1, 1, 9, 80

Offset: 1

Altug Alkan, Jul 29 2016

nonn

a(A057922(n)) = 1. - Michel Marcus, Aug 01 2016

			a(10) = 4 because A000005(10) divides A000005(10+4).

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A057922, A171937, A275478.

a(n) = {my(k = 1); while(numdiv(n+k) % numdiv(n) != 0, k++); k; }

Data section extended up to a(100) by Antti Karttunen, Mar 02 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A377319 a(n) is the smallest positive integer k such that n + k and n - k have the same number of divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A171936 Backwards van Eck transform of A000005.

Views

Author

Keywords

Links

Crossrefs

A275479 Least positive k such that d(n) divides d(n+k) (d = A000005).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions