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

1,1

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

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A024619, A089352.

Sequence in context: A305613 A267967 A051283 * A212666 A291046 A071140

Adjacent sequences: A066028 A066029 A066030 * A066032 A066033 A066034

Keyword

nonn

Author

Joseph L. Pe, Dec 12 2001

Extensions

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

Status

approved