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Number of iterations of x -> A003961(x) needed until A003415(x) <= x, when starting from x=n, where A003415(x) gives the arithmetic derivative of x, and A003961 shifts its prime factorization one step towards the larger primes.
4

%I #12 May 23 2021 03:20:54

%S 0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,2,0,1,0,1,0,0,0,2,0,0,0,1,0,1,0,2,0,0,

%T 0,2,0,0,0,2,0,0,0,1,0,0,0,2,0,0,0,1,0,1,0,2,0,0,0,2,0,0,0,3,0,0,0,1,

%U 0,0,0,2,0,0,0,1,0,0,0,2,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,3,0,0,0,1,0,0,0,2,0

%N Number of iterations of x -> A003961(x) needed until A003415(x) <= x, when starting from x=n, where A003415(x) gives the arithmetic derivative of x, and A003961 shifts its prime factorization one step towards the larger primes.

%H Antti Karttunen, <a href="/A343222/b343222.txt">Table of n, a(n) for n = 1..65537</a>

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A343222(n) = if(A003415(n)<=n,0,1+A343222(A003961(n)));

%Y Cf. A003415, A003961.

%Y Positions of zeros: Union of A051674 and A083347.

%Y Cf. also A343221, A344027.

%K nonn

%O 1,16

%A _Antti Karttunen_, Apr 08 2021