A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2160, 2240, 2250

Offset: 1

Author

Labos Elemer, May 13 2002

Keywords

• nonn

Comments

a(n) are the numbers such that the difference between the largest and the smallest prime divisor equals the sum of the other distinct prime divisors. - Michel Lagneau, Nov 13 2011

The statement above is only true for 966 of the first 1000 terms. The first counterexample is a(140) = 15015. - Donovan Johnson, Apr 10 2013

Lagneau's definition can be simplified to the largest prime divisor equals the sum of the other distinct prime divisors. - Christian N. K. Anderson, Apr 15 2013

Examples

			n = 70 = 2*5*7 has a form of 2pq, where p and q are twin primes; n = 3135 = 3*5*11*19, sum = 3+5+11+19 = 38 = 2*19, divisible by 19.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Subsequence of A024619.

Programs

Haskell

a071140 n = a071140_list !! (n-1)
a071140_list = filter (\x -> a008472 x `mod` a006530 x == 0) a024619_list
-- Reinhard Zumkeller, Apr 18 2013

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[s, 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 2250, And[Length@ # > 1, Divisible[Total@ #, Last@ #]] &[FactorInteger[#][[All, 1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

Formula

A008472(n)/A006530(n) is an integer and n has at least 3 distinct prime factors.

A008472(a(n)) mod A006530(a(n)) = 0 and A010055(a(n)) = 0. - Reinhard Zumkeller, Apr 18 2013