Sum of elements in the modified prime factorization of n: a(n) = 1 + sum p_i^e_i for n = p_1^e_1 * ... * p_k^e_k.
2, 3, 4, 5, 6, 6, 8, 9, 10, 8, 12, 8, 14, 10, 9, 17, 18, 12, 20, 10, 11, 14, 24, 8, 26, 16, 10, 12, 30, 8
1,1
For n > 1 with prime factorization n = p_1^e_1 * p_2^e_2 * ... * p_k^e_k, a(n) = 1 + sum_{i=1..k} p_i^e_i. For n=1, a(1) = 1 + 1 = 2.
Fixed point: a(6) = 6 is the unique fixed point.
Cycle: a(8) = 9, a(9) = 10, a(10) = 8 forms a 3-cycle.
Empirical observation: Iterating a(n) on any positive integer n eventually reaches either the fixed point 6 or the 3-cycle {8, 9, 10}.
a(n) = 1 + sum_{i=1..k} p_i^(e_i) for n = p_1^(e_1) * p_2^(e_2) * ... * p_k^(e_k) with p_i prime and e_i >= 1.
a(1) = 2.
a(8) = 1 + 2^3 = 9.
a(10) = 1 + 2^1 + 5^1 = 8.
Iteration example: 19018 -> 92 -> 28 -> 12 -> 8 -> 9 -> 10 -> 8.
(Python)
def a(n):
if n == 1: return 2
factors = []
d = 2
temp = n
while d * d <= temp:
if temp % d == 0:
power = 1
while temp % d == 0:
power *= d
temp //= d
factors.append(power)
d += 1
if temp > 1:
factors.append(temp)
return 1 + sum(factors)
nonn,changed
recycled
Jo Geon Woo, Aug 31 2026
