

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.


3



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
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OFFSET

1,4


LINKS

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


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)]


EXAMPLE

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.


PROG

(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); };


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.
Sequence in context: A101322 A029644 A024919 * A328099 A003415 A302055
Adjacent sequences: A328382 A328383 A328384 * A328386 A328387 A328388


KEYWORD

nonn


AUTHOR

Antti Karttunen, Oct 14 2019


STATUS

approved



