A328385 If n is of the form p^p, a(n) = n, otherwise a(n) is the first number found by iterating the map x -> A003415(x) that is different from n and either a prime, or whose degree (A051903) differs from the degree of n.

0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 7, 13, 1, 44, 10, 8, 27, 32, 1, 31, 1, 80, 9, 19, 12, 96, 1, 7, 16, 68, 1, 41, 1, 48, 39, 25, 1, 608, 14, 39, 20, 56, 1, 81, 16, 92, 13, 31, 1, 96, 1, 9, 51, 640, 18, 61, 1, 72, 8, 59, 1, 156, 1, 16, 55, 80, 18, 71, 1, 3424, 108, 43, 1, 128, 13, 45, 32, 140, 1, 123, 20, 96, 19

Offset: 1

Author

Antti Karttunen, Oct 14 2019

Keywords

• nonn

Examples

			For n = 3, 3 is a prime, thus a(3) = 1.
For n = 4, A003415(4) = 4, thus as it is among the fixed points of A003415 and a(4) = 4.
For n = 8 = 2^3, its "degree" is A051903(33) = 3, but A003415(8) = 12 = 2^2 * 3, with degree 2, thus a(8) = 12.
For n = 21 = 3*7, A051903(21) = 1, the first derivative A003415(21) = 10 = 2*5 is of the same degree as A051903(10) = 1, but then continuing, we have A003415(10) = 7, which is a prime, thus a(21) = 7.
For n = 33 = 3*11, A051903(33) = 1, A003415(33) = 14 = 2*7, is of the same degree, but on the second iteration, A003415(14) = 9 = 3^2, with A051903(9) = 2, different from the initial degree, thus a(33) = 9.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A003415, A051674, A051903, A157037.

Cf. A328384 (the number of iterations needed to reach such a number).

Cf. also A327968, A327975, A327977, A328252, A328305, A328310, A328311, A328320, A328321.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328385(n) = { my(d=A051903(n), u=A003415(n)); while(u && (u!=n) && !isprime(u) && A051903(u)==d, n = u; u = A003415(u)); (u); };

Formula

a(1) = 0 [as here the degrees of 0 and 1 are considered different].

a(p) = 1 for all primes.

a(A051674(n)) = A051674(n).

a(A157037(n)) = A003415(A157037(n)), a prime.

a(A328252(n)) = A003415(A328252(n)), a squarefree number.

a(n) = A003415^(k)(n), when k = abs(A328384(n)). [Taking the abs(A328384(n))-th arithmetic derivative of n gives a(n)]