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%I #27 Dec 20 2021 18:48:54
%S 30,60,90,120,150,180,240,270,300,360,450,480,540,600,720,750,810,858,
%T 900,960,1080,1200,1350,1440,1500,1620,1716,1722,1800,1920,2160,2250,
%U 2400,2430,2574,2700,2880,3000,3240,3432,3444,3600,3750,3840,4050,4320,4500
%N Numbers k such that 1/(Sum_{p|k} (1/p) - 1), where p are the prime divisors of k, is a positive integer.
%C 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.
%C 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
%H Amiram Eldar, <a href="/A069819/b069819.txt">Table of n, a(n) for n = 1..4233</a> (terms below 10^10)
%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/3261-a1862-inversons-les-facteurs">A1862 - Inversons les facteurs</a> (in French).
%e 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.
%t Select[Range[4320], (sum = Plus @@ (1/FactorInteger[#][[;;,1]])) > 1 && IntegerQ[1/(sum - 1)] &] (* _Amiram Eldar_, Feb 03 2020 *)
%o (Python)
%o from sympy import factorint
%o from fractions import Fraction
%o def ok(n):
%o s = sum(Fraction(1, p) for p in factorint(n))
%o return s > 1 and (s - 1).numerator == 1
%o print([k for k in range(1, 4501) if ok(k)]) # _Michael S. Branicky_, Dec 19 2021
%o (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
%Y Cf. A007850.
%Y A143207 is a subsequence.
%K easy,nonn
%O 1,1
%A _Benoit Cloitre_, Apr 28 2002
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