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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.

%I A071140 #26 Jan 10 2025 10:08:33
%S A071140 30,60,70,90,120,140,150,180,240,270,280,286,300,350,360,450,480,490,
%T A071140 540,560,572,600,646,700,720,750,810,900,960,980,1080,1120,1144,1200,
%U A071140 1292,1350,1400,1440,1500,1620,1750,1798,1800,1920,1960,2160,2240,2250
%N 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.
%C A071140 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
%C A071140 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
%C A071140 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
%H A071140 Donovan Johnson, <a href="/A071140/b071140.txt">Table of n, a(n) for n = 1..1000</a>
%F A071140 A008472(n)/A006530(n) is an integer and n has at least 3 distinct prime factors.
%F A071140 A008472(a(n)) mod A006530(a(n)) = 0 and A010055(a(n)) = 0. - _Reinhard Zumkeller_, Apr 18 2013
%e A071140 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.
%t A071140 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}]
%t A071140 (* Second program: *)
%t A071140 Select[Range@ 2250, And[Length@ # > 1, Divisible[Total@ #, Last@ #]] &[FactorInteger[#][[All, 1]] ] &] (* _Michael De Vlieger_, Jul 18 2017 *)
%o A071140 (Haskell)
%o A071140 a071140 n = a071140_list !! (n-1)
%o A071140 a071140_list = filter (\x -> a008472 x `mod` a006530 x == 0) a024619_list
%o A071140 -- _Reinhard Zumkeller_, Apr 18 2013
%Y A071140 Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.
%Y A071140 Subsequence of A024619.
%K A071140 nonn
%O A071140 1,1
%A A071140 _Labos Elemer_, May 13 2002