A336099 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.

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, 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, 1, 5, 1, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 2

Author

Vladimir Letsko, Jul 08 2020

Keywords

• nonn

Comments

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

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.

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

Examples

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

Links

• Vladimir Letsko, Mathematical Marathon, Problem 227 (in Russian).

Crossrefs

Cf. A008472, A336098, A336296.

Cf. A158804 (all possible k's).