A097246 Replace factors of n that are squares of a prime with the prime succeeding this prime.

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 9, 13, 14, 15, 9, 17, 10, 19, 15, 21, 22, 23, 18, 7, 26, 15, 21, 29, 30, 31, 18, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 25, 46, 47, 27, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 45, 61, 62, 35, 27, 65, 66, 67, 51, 69, 70, 71, 30, 73

Offset: 1

Author

Reinhard Zumkeller, Aug 03 2004

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000035, A000040, A002110, A049084, A097250, A000079, A000188, A000244, A003961, A004526, A005117, A007814, A007913, A048675, A064989, A151800.

Cf. A097247, A097248 (fixed points of iteration), A097249 (number of iterations needed to reach them for each n), A277886, A277899.

Programs

Mathematica

Table[Times @@ Map[#1^#2 & @@ # &, Partition[#, 2, 2] &@ Flatten[ FactorInteger[n] /. {p_, e_} /; e >= 2 :> {If[OddQ@ e, {p, 1}, {1, 1}], {NextPrime@ p, Floor[e/2]}}]], {n, 73}] (* 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))); }; \\ Antti Karttunen, Mar 18 2017

Python

from sympy import factorint, nextprime
from operator import mul
def a(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]) # Indranil Ghosh, May 15 2017

Scheme

(definec (A097246 n) (if (= 1 n) 1 (* (A000244 (A004526 (A007814 n))) (A000079 (A000035 (A007814 n))) (A003961 (A097246 (A064989 n))))))
(define (A097246 n) (* (A003961 (A000188 n)) (A007913 n)))
;; Antti Karttunen, Nov 15 2016

Formula

Multiplicative with p^e -> NextPrime(p)^floor(e/2) * p^(e mod 2), where p prime and NextPrime(p)=A000040(A049084(p)+1).

a(n) <= n; a(n) = n iff n is squarefree: a(A005117(n)) = A005117(n);

a(m*n) <= a(m)*a(n); a(m*n) = a(m)*a(n) iff m and n are coprime;

a(A000040(k)^n) = A000040(k+1)^floor(n/2)*A000040(k)^(n mod 2); a(2^n) = 3^floor(n/2) * (1 + n mod 2);

a(A000040(k)*A002110(n)/A002110(k-1)) = A000040(k+1)*A002110(n)/A002110(k) for k <= n, see also A097250.

From Antti Karttunen, Nov 15 2016: (Start)

a(1) = 1; for n > 1, a(n) = 2^A000035(A007814(n)) * 3^A004526(A007814(n)) * A003961(a(A064989(n))).

a(n) = A003961(A000188(n)) * A007913(n).

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

(End)

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (p^4-p^2)/(p^4-nextprime(p)) = 0.4059779303..., where nextprime is A151800. - Amiram Eldar, Nov 29 2022