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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.
3
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
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. A158804 (all possible k's).
Sequence in context: A305052 A305079 A319841 * A290090 A066075 A359211
KEYWORD
nonn
AUTHOR
Vladimir Letsko, Jul 08 2020
STATUS
approved