A377482 - OEIS

%I #6 Nov 19 2024 02:25:37

%S 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,

%T 20,11,29,17,31,17,34,19,19,17,37,38,35,11,41,19,43,34,11,42,47,11,34,

%U 19,40,17,53,11,35,13,35,31,59,19,61,67,13,19,37,35,67

%N 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.

%C 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.

%e 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.

%o (Python)

%o from sympy import*

%o def a(n):

%o  t=0

%o  while n not in (1,4) and not isprime(n):

%o   n=sum(p*e for p,e in factorint(n).items());t+=n

%o  return t or n

%Y Cf. A001414, A274718.

%K nonn,easy,look

%O 1,2

%A _Louis-Simon Cyr_, Oct 29 2024