A064802 a(n) = Min { m > n | prime factorizations of m and n differ in one factor only}, a(1) = 1.

1, 3, 5, 6, 7, 9, 11, 12, 15, 14, 13, 18, 17, 21, 21, 24, 19, 27, 23, 28, 33, 26, 29, 36, 35, 34, 45, 42, 31, 42, 37, 48, 39, 38, 49, 54, 41, 46, 51, 56, 43, 63, 47, 52, 63, 58, 53, 72, 77, 70, 57, 68, 59, 81, 65, 84, 69, 62, 61, 84, 67, 74, 99, 96, 85, 78, 71, 76, 87, 98, 73

Offset: 1

Author

Reinhard Zumkeller, Oct 21 2001

Keywords

• nonn
• easy

Comments

a(A000040(k)) = A000040(k + 1).

A094457 gives next smaller comparable number, replacing the prime factor 2 with 1. - Michael De Vlieger, Jan 31 2015

From Peter Munn, Oct 13 2023: (Start)

For n > 1, a(n) is the smallest number m > n in the factorization neighborhood of n given by A127185(m, n) <= 2.

Usually, the minimum m is achieved by replacing the largest prime factor with the next prime. So through the first 60 terms about 1 term in 5 differs from the corresponding term of A253550, but this proportion drops to 611 of the first 10000 terms. Nevertheless, I see reasons (deriving from the distribution of the lengths of prime gaps) to doubt that the asymptotic density of {n : a(n) <> A253550(n)} is less than 611/10000.

(End)

Examples

			n = 20 = 2 * 2 * 5: as 2 * 3 * 5 > 2 * 2 * 7 = 28 we have a(20) = 28.
		

Links

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

Crossrefs

Cf. A000040, A094457, A127185, A253550.

Programs

Mathematica

f[n_] := Block[{g}, g[x_] := Flatten[Table[#1, {#2}] & @@@ FactorInteger@ x]; If[n == 1, 1, Min[Times @@ MapAt[NextPrime, g[n], #] & /@ Range[Length@ g[n]]]]]; Array[f, 71] (* Michael De Vlieger, Jan 31 2015 *)