cp's OEIS Frontend

A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.

%I A071139 #19 Jan 10 2025 04:30:04
%S A071139 2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,30,31,32,37,41,43,47,49,53,
%T A071139 59,60,61,64,67,70,71,73,79,81,83,89,90,97,101,103,107,109,113,120,
%U A071139 121,125,127,128,131,137,139,140,149,150,151,157,163,167,169,173,179,180
%N A071139 Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k.
%C A071139 All primes and prime powers are terms, as are certain other composites (see Example section).
%C A071139 If k is a term then every multiple of k having no prime factors other than those of k are also terms. E.g., since 286 = 2*11*13 is a term, so are 572 = 286*2 and 3146 = 286*11.
%C A071139 If k = 2*p*q where p and q are twin primes, then sum = 2+p+q = 2q is divisible by q, the largest prime factor, so 2*A037074 is a subsequence.
%H A071139 Reinhard Zumkeller, <a href="/A071139/b071139.txt">Table of n, a(n) for n = 1..10000</a>
%F A071139 A008472(k)/A006530(k) is an integer.
%e A071139 30 = 2*3*5; sum of distinct prime factors is 2+3+5 = 10, which is divisible by 5, so 30 is a term;
%e A071139 2181270 = 2*3*5*7*13*17*47; sum of distinct prime factors is 2+3+5+7+13+17+47 = 94, which is divisible by 47, so 2181270 is a term.
%t A071139 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], Print[{n, ba[n]}]], {n, 2, 1000000}]
%o A071139 (Haskell)
%o A071139 a071139 n = a071139_list !! (n-1)
%o A071139 a071139_list = filter (\x -> a008472 x `mod` a006530 x == 0) [2..]
%o A071139 -- _Reinhard Zumkeller_, Apr 18 2013
%o A071139 (PARI) isok(n) = if (n != 1, my(f=factor(n)[,1]); (sum(k=1, #f~, f[k]) % vecmax(f)) == 0); \\ _Michel Marcus_, Jul 09 2018
%Y A071139 Cf. A008472, A006530, A000961, A025475, A037074.
%K A071139 nonn
%O A071139 1,1
%A A071139 _Labos Elemer_, May 13 2002
%E A071139 Edited by _Jon E. Schoenfield_, Jul 08 2018