A377482 Iterated integer log of n: denote A001414(n) by b(n). a(n) = n if b(n) = n. Otherwise, a(n) = b(n) + b(b(n)) + ... + b^k(n), where k is the smallest integer such that b^k(n) is prime.

1, 2, 3, 4, 5, 5, 7, 11, 11, 7, 11, 7, 13, 20, 19, 19, 17, 19, 19, 20, 17, 13, 23, 20, 17, 34, 20, 11, 29, 17, 31, 17, 34, 19, 19, 17, 37, 38, 35, 11, 41, 19, 43, 34, 11, 42, 47, 11, 34, 19, 40, 17, 53, 11, 35, 13, 35, 31, 59, 19, 61, 67, 13, 19, 37, 35, 67

Offset: 1

Author

Louis-Simon Cyr, Oct 29 2024

Keywords

• nonn
• easy
• look

Comments

Can be understood as an exotic way of measuring how far a number is from being prime, since omitting n = 1, 4, |a(n) - n| = 0 if and only if n is prime. Note that A274718(n) = k - 1 when a(n) = b(n) + b(b(n)) + ... + b^k(n). The scatter plot for n >= 10000 shows intriguing regularities.

Examples

			a(24) is computed as follows: 24 = (2^3) * 3, 2 * 3 + 3 = 9. 9 = (3^2), 3 + 3 = 6. 6 = 2 * 3, 2 + 3 = 5. Since 5 is prime, we stop and take the sum: 9 + 6 + 5 = 20.
		

Crossrefs

Cf. A001414, A274718.

Programs

Python

from sympy import*
def a(n):
 t=0
 while n not in (1,4) and not isprime(n):
  n=sum(p*e for p,e in factorint(n).items());t+=n
 return t or n