A064802 - OEIS

%I #17 Oct 14 2023 14:28:47

%S 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,

%T 34,45,42,31,42,37,48,39,38,49,54,41,46,51,56,43,63,47,52,63,58,53,72,

%U 77,70,57,68,59,81,65,84,69,62,61,84,67,74,99,96,85,78,71,76,87,98,73

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

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

%C A094457 gives next smaller comparable number, replacing the prime factor 2 with 1. - _Michael De Vlieger_, Jan 31 2015

%C From _Peter Munn_, Oct 13 2023: (Start)

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

%C 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.

%C (End)

%H Michael De Vlieger, <a href="/A064802/b064802.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%Y Cf. A000040, A094457, A127185, A253550.

%K nonn,easy

%O 1,2

%A _Reinhard Zumkeller_, Oct 21 2001