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

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, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167, 169, 173, 179, 180

Offset: 1

Author

Labos Elemer, May 13 2002

Keywords

• nonn

Comments

All primes and prime powers are terms, as are certain other composites (see Example section).

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.

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.

Examples

			30 = 2*3*5; sum of distinct prime factors is 2+3+5 = 10, which is divisible by 5, so 30 is a term;
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.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074.

Programs

Haskell

a071139 n = a071139_list !! (n-1)
a071139_list = filter (\x -> a008472 x `mod` a006530 x == 0) [2..]
-- 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], Print[{n, ba[n]}]], {n, 2, 1000000}]

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

Formula

A008472(k)/A006530(k) is an integer.

Extensions

Edited by Jon E. Schoenfield, Jul 08 2018