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).

1, 5, 19, 21, 154, 604

Offset: 1

Author

Paolo P. Lava, May 29 2014

Keywords

• nonn
• more

Comments

a(7) > 10^8. - Michel Marcus, Nov 01 2014

a(7), if it exists, is > 10^9. - Vaclav Kotesovec, Oct 16 2018

Examples

			phi(1) + phi(2) + phi(3) + phi(4) + phi(5) = 1 + 1 + 2 + 2 + 4 = 10;
d(1) + d(2) + d(3) + d(4) + d(5) = 1 + 2 + 2 + 3 + 2 = 10;
Finally 10 / 10 = 1.
		

Crossrefs

Cf. A000005, A000010, A002088, A006218, A226647.

Programs

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

Maple

with(numtheory):P:=proc(q) local a,b,n; a:=0; b:=0;
for n from 1 to q do a:=a+tau(n); b:=b+phi(n);
if type(b/a,integer) then print(n); fi; od; end: P(10^10);

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

Formula

Numbers k such that A006218(k) divides A002088(k).