A066031 Composite numbers n the sum of whose prime factors divides n, but which are not themselves powers of primes.

30, 60, 70, 84, 90, 105, 120, 140, 150, 168, 180, 231, 234, 240, 252, 260, 270, 280, 286, 300, 315, 336, 350, 360, 450, 456, 468, 480, 490, 504, 520, 525, 528, 532, 540, 560, 572, 588, 600, 627, 646, 672, 693, 700, 702, 720, 735, 750, 756, 805, 810, 897

Offset: 1

Author

Joseph L. Pe, Dec 12 2001

Keywords

• nonn

Comments

Primes and powers of primes have been excluded from the sequence because they trivially satisfy the condition "the sum of the prime factors of n divides n". Call a term of the sequence "primitive" if it is not a multiple of some previous term; for example, 70 is primitive while 60 is not. Are there infinitely many primitive terms? See A064623.

Intersection of A089352 and A024619. - Michel Marcus, Feb 03 2016

Examples

			The sum of the prime factors of 70 is 2 + 5 + 7 = 14, which divides 70.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Jean-Marie de Koninck, Florian Luca, Integers divisible by sums of powers of their prime factors, Journal of Number Theory, Volume 128, Issue 3, March 2008, Pages 557-563.
• J. Pe, The Prime-perfect Numbers

Crossrefs

Cf. A024619, A089352.

Programs

Mathematica

Select[ Range[2, 900], IntegerQ[ # / Apply[ Plus, First[ Transpose[ FactorInteger[ # ]]]]] && Mod[ #, # - EulerPhi[ # ]] != 0 & ]

PARI

isok(n) = if (omega(n)<2, return(0)); my(f = factor(n)) ; (n % vecsum(f[,1])) == 0; \\ Michel Marcus, Feb 03 2016

Extensions

More terms from Robert G. Wilson v, Dec 12 2001