cp's OEIS Frontend

A243018 Numbers k such that Sum_{i=1..k} phi(i) is divisible by Sum_{i=1..k} d(i), where phi(i) is the Euler totient function of i (A000010), and d(i) is the number of divisors of i (A000005).

%I A243018 #23 Sep 08 2022 08:46:08
%S A243018 1,5,19,21,154,604
%N A243018 Numbers k such that Sum_{i=1..k} phi(i) is divisible by Sum_{i=1..k} d(i), where phi(i) is the Euler totient function of i (A000010), and d(i) is the number of divisors of i (A000005).
%C A243018 a(7) > 10^8. - _Michel Marcus_, Nov 01 2014
%C A243018 a(7), if it exists, is > 10^9. - _Vaclav Kotesovec_, Oct 16 2018
%F A243018 Numbers k such that A006218(k) divides A002088(k).
%e A243018 phi(1) + phi(2) + phi(3) + phi(4) + phi(5) = 1 + 1 + 2 + 2 + 4 = 10;
%e A243018 d(1) + d(2) + d(3) + d(4) + d(5) = 1 + 2 + 2 + 3 + 2 = 10;
%e A243018 Finally 10 / 10 = 1.
%p A243018 with(numtheory):P:=proc(q) local a,b,n; a:=0; b:=0;
%p A243018 for n from 1 to q do a:=a+tau(n); b:=b+phi(n);
%p A243018 if type(b/a,integer) then print(n); fi; od; end: P(10^10);
%o A243018 (PARI) lista(nn) = {se = 0; sn = 0; for (n=1, nn, se += eulerphi(n); sn += numdiv(n); if (se % sn == 0, print1(n, ", ")););} \\ _Michel Marcus_, Nov 01 2014
%o A243018 (Magma) [n:n in [1..1000]| IsIntegral(&+[EulerPhi(m):m in [1..n]]/&+[NumberOfDivisors(m):m in [1..n]])] ; // _Marius A. Burtea_, Mar 25 2019
%Y A243018 Cf. A000005, A000010, A002088, A006218, A226647.
%K A243018 nonn,more
%O A243018 1,2
%A A243018 _Paolo P. Lava_, May 29 2014