cp's OEIS Frontend

A082300 Numbers relatively prime to the sum of their prime factors (with repetition).

%I A082300 #31 Apr 16 2025 04:30:34
%S A082300 1,6,10,12,14,15,20,21,22,26,28,33,34,35,38,39,40,44,45,46,48,51,52,
%T A082300 54,55,56,57,58,62,63,65,68,69,74,75,76,77,80,82,85,86,87,88,90,91,92,
%U A082300 93,94,95,96,99,104,106,108,111,112,115,116,117,118,119,122,123,124,129,133
%N A082300 Numbers relatively prime to the sum of their prime factors (with repetition).
%C A082300 In other words, numbers n such that n and sopfr(n) are relatively prime, where sopfr(n) (A001414) is the sum of the primes (with repetition) dividing n.
%C A082300 Conjecture: a(n) ~ (Pi^2/6)n. - _Charles R Greathouse IV_, Aug 04 2016
%C A082300 No term is prime since for prime p, p and 2p are not coprime; similarly no term is a prime power. A050703 is a subsequence because then n+sopfr(n) is prime, and so coprime to n. - _David James Sycamore_, Mar 04 2018
%H A082300 Charles R Greathouse IV, <a href="/A082300/b082300.txt">Table of n, a(n) for n = 1..10000</a>
%e A082300 gcd(2*2*5,2+2+5) = gcd(2*2*5,3*3)=1, therefore 20 is a term;
%e A082300 gcd(3*11,3+11) = gcd(3*11,2*7)=1, therefore 33 is a term.
%t A082300 Select[Range@ 106, CoprimeQ[#, Total@ Flatten@ Map[Table[#1, {#2}] & @@ # &, FactorInteger[#]]] &] (* _Michael De Vlieger_, Aug 06 2016 *)
%o A082300 (PARI) sopfr(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]*f[i,2])
%o A082300 is(n)=gcd(sopfr(n),n)==1 \\ _Charles R Greathouse IV_, Aug 04 2016
%Y A082300 Cf. A001414, A275665, A050703.
%Y A082300 A082299(a(n)) = 1.
%K A082300 nonn
%O A082300 1,2
%A A082300 _Reinhard Zumkeller_, Apr 08 2003
%E A082300 Revised definition from _Lior Manor_, Apr 14 2004