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A069819
Numbers k such that 1/(Sum_{p|k} (1/p) - 1), where p are the prime divisors of k, is a positive integer.
3
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
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Apr 28 2002
STATUS
approved