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Number of solutions of the equation k = n*sopf(k) in positive integers where sopf(k) is the sum of distinct prime factors of k.
3

%I #39 Dec 10 2023 22:05:01

%S 1,2,1,2,1,3,1,2,1,3,1,3,1,1,1,2,1,4,1,2,1,3,1,2,2,2,1,2,1,4,1,2,2,2,

%T 1,3,2,1,1,2,1,4,1,1,0,3,1,3,1,1,2,2,1,0,1,2,2,4,1,1,2,2,1,1,1,4,2,1,

%U 1,5,1,2,2,1,2,1,1,2,1

%N Number of solutions of the equation k = n*sopf(k) in positive integers where sopf(k) is the sum of distinct prime factors of k.

%C Offset is 2 because a(1) cannot be defined since there are infinitely many solutions for n = 1, the primes.

%C If n = p^s then p^(s+1) is solution of k = n*sopf(k). Hence a(p^s) > 0. On the other hand there are infinitely many 0's in the sequence. For example a(5^s*11^t) = 0 for all positive integers s, t.

%C Records appear to occur only at prime n. These are seen in A336296, although note that A336296 is not monotonic, so it includes other terms. - _Bill McEachen_, Dec 02 2023

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

%e a(3) = 2 because there are exactly 2 solutions of the equation k = 3*sopf(k) in positive integers (9 and 30).

%Y Cf. A008472, A336098, A336296.

%Y Cf. A158804 (all possible k's).

%K nonn

%O 2,2

%A _Vladimir Letsko_, Jul 08 2020