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

2,2

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

Links

Table of n, a(n) for n=2..80.

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

Example

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

Crossrefs

Cf. A008472, A336098, A336296.

Cf. A158804 (all possible k's).

Sequence in context: A305052 A305079 A319841 * A290090 A066075 A359211

Adjacent sequences: A336096 A336097 A336098 * A336100 A336101 A336102

Keyword

nonn

Author

Vladimir Letsko, Jul 08 2020

Status

approved