A097248 a(n) is the eventual stable point reached when iterating k -> A097246(k), starting from k = n.

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 5, 13, 14, 15, 5, 17, 10, 19, 15, 21, 22, 23, 10, 7, 26, 15, 21, 29, 30, 31, 10, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 7, 46, 47, 15, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 7, 61, 62, 35, 15, 65, 66, 67, 51, 69, 70, 71, 30, 73, 74, 21

Offset: 1

Author

Reinhard Zumkeller, Aug 03 2004

Keywords

• nonn

Comments

a(n) = r(n,m) with m such that r(n,m)=r(n,m+1), where r(n,k) = A097246(r(n,k-1)), r(n,0)=n. (The original definition.)

A097248(n) = r(n,a(n)).

From Antti Karttunen, Nov 15 2016: (Start)

The above remark could be interpreted to mean that A097249(n) <= a(n).

All terms are squarefree, and the squarefree numbers are the fixed points.

These are also fixed points eventually reached when iterating A277886.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Range of values is A005117.

Cf. A008683, A019565, A048675, A097246, A097247, A097249, A277905, A332824.

A003961, A225546, A277885, A277886, A331590 are used to express relationship between terms of this sequence.

The formula section also details how the sequence maps the terms of A007913, A260443, A329050, A329332.

See comments/formulas in A283475, A283478, A331751 giving their relationship to this sequence.

Programs

Mathematica

Table[FixedPoint[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[FactorInteger[#] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]] &, n], {n, 75}] (* Michael De Vlieger, Mar 18 2017 *)

PARI

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
\\ Antti Karttunen, Mar 18 2017

Python

from sympy import factorint, nextprime
from operator import mul
def a097246(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(nextprime(i)**int(f[i]/2))*(i**(f[i]%2)) for i in f])
def a(n):
    k=a097246(n)
    while k!=n:
        n=k
        k=a097246(k)
    return k # Indranil Ghosh, May 15 2017

Scheme

;; with memoization-macro definec
;; Two implementations:
(definec (A097248 n) (if (not (zero? (A008683 n))) n (A097248 (A097246 n))))
(definec (A097248 n) (if (zero? (A277885 n)) n (A097248 (A277886 n))))
;; Antti Karttunen, Nov 15 2016

Formula

a(A005117(n)) = A005117(n).

From Antti Karttunen, Nov 15 2016: (Start)

If A008683(n) <> 0 [when n is squarefree], a(n) = n, otherwise a(n) = a(A097246(n)).

If A277885(n) = 0, a(n) = n, otherwise a(n) = a(A277886(n)).

A007913(a(n)) = a(n).

a(A007913(n)) = A007913(n).

A048675(a(n)) = A048675(n).

a(A260443(n)) = A019565(n).

(End)

From Peter Munn, Feb 06 2020: (Start)

a(1) = 1; a(p) = p, for prime p; a(m*k) = A331590(a(m), a(k)).

a(A331590(m,k)) = A331590(a(m), a(k)).

a(n^2) = a(A003961(n)) = A003961(a(n)).

a(A225546(n)) = a(n).

a(n) = A225546(2^A048675(n)) = A019565(A048675(n)).

a(A329050(n,k)) = prime(n+k-1) = A000040(n+k-1).

a(A329332(n,k)) = A019565(n * k).

Equivalently, a(A019565(n)^k) = A019565(n * k).

(End)

From Antti Karttunen, Feb 22-25 & Mar 01 2020: (Start)

a(A019565(x)*A019565(y)) = A019565(x+y).

a(A332461(n)) = A332462(n).

a(A332824(n)) = A019565(n).

a(A277905(n,k)) = A277905(n,1) = A019565(n), for all n >= 1, and 1 <= k <= A018819(n).

(End)

Extensions

Name changed and the original definition moved to the Comments section by Antti Karttunen, Nov 15 2016