A158804 Composite integers that are a multiple of the sum of their distinct prime factors.

4, 8, 9, 16, 25, 27, 30, 32, 49, 60, 64, 70, 81, 84, 90, 105, 120, 121, 125, 128, 140, 150, 168, 169, 180, 231, 234, 240, 243, 252, 256, 260, 270, 280, 286, 289, 300, 315, 336, 343, 350, 360, 361, 450, 456, 468, 480, 490, 504, 512, 520, 525, 528, 529, 532, 540

Offset: 1

Author

R. J. Mathar, Mar 27 2009

Keywords

• easy
• nonn

Comments

Koninck & Luca give upper and lower bounds for the number of elements of this sequence below x: x / exp(c_i(1 + o(1))sqrt(log x log log x)), where the constants c_i and the o(1) differ for lower and upper bounds. - Charles R Greathouse IV, Sep 08 2012

Examples

			4 is in the sequence because A008472(4)=2 divides 4. 5 is not in the sequence because it is prime. 6 is not in the sequence because A008472(6)=5 does not divide 6.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Jean-Marie de Koninck, Florian Luca, Integers divisible by the sum of their prime factors, Mathematika 52 (1&2) (2005) 69-77, MR2261843.

Crossrefs

Cf. A002808 (composite numbers), A008472.

Programs

Maple

A008472 := proc(n) numtheory[factorset](n) ; add(d,d=%) ; end: isbeta := proc(n) if isprime(n) then false; else if n mod A008472(n) = 0 then true; else false; fi; fi; end: for n from 2 to 1200 do if isbeta(n) then printf("%d,",n); fi; od:

Mathematica

Select[Range[2,540],!PrimeQ[#]&&IntegerQ[#/Total[First/@FactorInteger[#]]]&] (* Jayanta Basu, Jun 02 2013 *)

PARI

is(n)=my(f=factor(n)[,1]);n%sum(i=1,#f,f[i])==0 \\ Charles R Greathouse IV, Feb 04 2013

Formula

{n in A002808: A008472(n)|n }