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

Author

Antti Karttunen, Dec 15 2015

Keywords

• nonn
• easy
• mult

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

Crossrefs

Cf. A003586 (fixed points), A065331.

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

Programs

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 *)

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))))

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).

Extensions

Keyword mult added by Antti Karttunen, Aug 04 2018