A069819 Numbers k such that 1/(Sum_{p|k} (1/p) - 1), where p are the prime divisors of k, is a positive integer.

30, 60, 90, 120, 150, 180, 240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810, 858, 900, 960, 1080, 1200, 1350, 1440, 1500, 1620, 1716, 1722, 1800, 1920, 2160, 2250, 2400, 2430, 2574, 2700, 2880, 3000, 3240, 3432, 3444, 3600, 3750, 3840, 4050, 4320, 4500

Offset: 1

Author

Benoit Cloitre, Apr 28 2002

Keywords

• easy
• nonn

Comments

Sequence is generated by A007850(n). For example: 30, 858, 1722 (30 = 2*3*5, 858 = 2*3*11*13, 1722 = 2*3*11*13) generate numbers of the form 2^a*3^b*5^c (A143207), 2^a*3^b*7^c*41^d, 2^a*3^b*11^c*13^d, (a,b,c,d => 1), which are in the sequence.

Equivalently, numbers k such that Sum_{p|k} 1/p - Product_{p|k} 1/p, where p are the prime divisors of k, is a positive integer. All these terms have at least 3 prime factors. When k is a term and p is a prime divisor of k, then p*k is another term (see Diophante link). - Bernard Schott, Dec 19 2021

Examples

			For k = 30 = 2*3*5, 1/(Sum_{p|n} (1/p) - 1) = 1/(1/2 + 1/3 + 1/5 - 1) = 30 hence 30 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..4233 (terms below 10^10)
• Diophante, A1862 - Inversons les facteurs (in French).

Crossrefs

Cf. A007850.

A143207 is a subsequence.

Programs

Mathematica

Select[Range[4320], (sum = Plus @@ (1/FactorInteger[#][[;;,1]])) > 1 && IntegerQ[1/(sum - 1)] &] (* Amiram Eldar, Feb 03 2020 *)

PARI

isok(k) = my(f=factor(k), x=1/(sum(i=1, #f~, 1/f[i,1]) -1)); (x>1) && (denominator(x)==1); \\ Michel Marcus, Dec 19 2021

Python

from sympy import factorint
from fractions import Fraction
def ok(n):
    s = sum(Fraction(1, p) for p in factorint(n))
    return s > 1 and (s - 1).numerator == 1
print([k for k in range(1, 4501) if ok(k)]) # Michael S. Branicky, Dec 19 2021