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

1,1

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

Links

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

Formula

A008472(n)/A006530(n) is 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

Example

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.

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 *)

Prog

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

Crossrefs

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

Subsequence of A024619.

Sequence in context: A066031 A212666 A291046 * A221054 A365795 A074915

Adjacent sequences: A071137 A071138 A071139 * A071141 A071142 A071143

Keyword

nonn

Author

Labos Elemer, May 13 2002

Status

approved