A265399 Repeatedly perform x^2 -> x+1 reduction for polynomial (with nonnegative integer coefficients) encoded in prime factorization of n, until the polynomial is at most degree 1.

1, 2, 3, 4, 6, 6, 18, 8, 9, 12, 108, 12, 1944, 36, 18, 16, 209952, 18, 408146688, 24, 54, 216, 85691213438976, 24, 36, 3888, 27, 72, 34974584955819144511488, 36, 2997014624388697307377363936018956288, 32, 324, 419904, 108, 36, 104819342594514896999066634490728502944926883876041385836544, 816293376, 5832, 48

Offset

1,2

Comments

In terms of integers: apply A265398 as many times as necessary to n, until it gets 3-smooth, one of the terms of A003586.

Completely multiplicative with a(2) = 2, a(3) = 3, a(p) = a(A265398(p)) for p > 3. - Andrew Howroyd & Antti Karttunen, Aug 04 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..60

Formula

If A065331(n) = n [that is, when n is one of 3-smooth numbers, A003586] then a(n) = n, otherwise a(n) = a(A265398(n)).

Other identities. For all n >= 1:

a(n) = 2^A265752(n) * 3^A265753(n).

Mathematica

f[p_, e_] := If[p < 5, p, a[NextPrime[p, -1] * NextPrime[p, -2]]]^e; a[1] = 1; a[n_] := a[n] = Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Sep 07 2023 *)

Prog

(PARI)
\\ Needs also code from A265398.
A265399(n) = if(A065331(n) == n, n, A265399(A265398(n)));
for(n=1, 60, write("b265399.txt", n, " ", A265399(n)));
(Scheme) (definec (A265399 n) (if (= (A065331 n) n) n (A265399 (A265398 n))))

Crossrefs

Cf. A003586 (fixed points), A065331.

Cf. also A192232, A206296, A265398, A265752, A265753.

Sequence in context: A265398 A299438 A030209 * A138588 A344412 A143102

Adjacent sequences: A265396 A265397 A265398 * A265400 A265401 A265402

Keyword

nonn,easy,mult

Author

Antti Karttunen, Dec 15 2015

Extensions

Keyword mult added by Antti Karttunen, Aug 04 2018

Status

approved