A336098 - OEIS

%I #17 Jul 19 2020 07:00:43

%S 46,55,85,87,92,110,123,138,141,145,155,158,183,184,187,190,194,203,

%T 205,217,219,220,230,238,247,253,259,261,265,275,276,282,287,290,291,

%U 295,302,305,310,316,319,327,334,339,366,368,369,380,388,391,395,403,406,407,410,414,415,423,425,426,427,434

%N Numbers k such that equation x = k*sopf(x) has no solutions in positive integers.

%C If k = p^s then p^(s+1) is solution of x = k*sopf(x). Hence powers of primes are not in the sequence.

%C Let p_1*...*p_t is in the sequence. Then p_1^a_1*...*p_t^a_t is in the sequence for all positive integers a_1, ..., a_t. It means that the sequence is infinite.

%H Vladimir Letsko, <a href="https://dxdy.ru/post1257616.html#p1257616">Mathematical Marathon, Problem 227</a> (in Russian).

%o (PARI) sopf(n) = vecsum(factor(n)[, 1]); \\ A008472

%o pp(n) = prod(k=1, n, prime(k)); \\ A002110

%o sp(n) = sum(k=1, n, prime(k)); \\ A007504

%o ip(n) = {my(k=1); while (pp(k)/sp(k) <= n, k++); k+1;}

%o listako(nn) = {my(lim = pp(ip(nn))); my(v = vector(lim, k, k++; k/sopf(k))); my(w = vector(nn-1, k, #select(x->(x==k+1), v))); apply(x->(x+1), Vec(select(x->(x==0), w, 1)));} \\ _Michel Marcus_, Jul 16 2020

%Y Cf. A008472, A336099.

%K nonn

%O 1,1

%A _Vladimir Letsko_, Jul 08 2020