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

1,1

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..4233 (terms below 10^10)

Diophante, A1862 - Inversons les facteurs (in French).

Example

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.

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A007850.

A143207 is a subsequence.

Sequence in context: A249674 A050519 A358756 * A143207 A359410 A108454

Adjacent sequences: A069816 A069817 A069818 * A069820 A069821 A069822

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 28 2002

Status

approved