A243373 - OEIS

%I #35 Feb 14 2021 08:16:05

%S 1,2,6,9,10,14,18,26,42,66,90,126,150,186,234,266,342,490,666,1426,

%T 1634,2394,4410,12834,14706,16758,18846,209754,308602,350154,385434,

%U 1122786,2777418,12130734,15616986,29682342,223843466,270397974,300398714,559894482

%N Numbers m such that k*phi(n) = Sum_{j|n} sigma(j), where k >= 1 is an integer.

%C a(49) > 10^11. - _Hiroaki Yamanouchi_, Aug 24 2014

%H Hiroaki Yamanouchi, <a href="/A243373/b243373.txt">Table of n, a(n) for n = 1..48</a>

%e The divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 and sigma(1) + sigma(2) + sigma(3) + sigma(5) + sigma(6) + sigma(9) + sigma(10) + sigma(15) + sigma(18) + sigma(30) + sigma(45) + sigma(90) = 1 + 3 + 4 + 6 + 12 + 13 + 18 + 24 + 39 + 72 + 78 + 234 = 504 and phi(n) = 24. Finally 504 / 24 = 21.

%p with(numtheory): P:=proc(q) local a,b,k,n;

%p for n from 1 to q do a:=divisors(n); b:=0;

%p b:=add(sigma(a[k]), k=1..nops(a)); if type(b/phi(n),integer)

%p then print(n); fi; od; end: P(10^10);

%o (PARI) isok(n) = (sumdiv(n, d, sigma(d)) % eulerphi(n)) == 0; \\ _Michel Marcus_, Jun 04 2014

%Y Cf. A000010, A000203, A221219.

%K nonn

%O 1,2

%A _Paolo P. Lava_, Jun 04 2014

%E a(37)-a(40) from _Hiroaki Yamanouchi_, Aug 24 2014