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

1,2

Links

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

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

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 *)

Prog

(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
(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
(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

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.

Sequence in context: A343247 A097248 A097247 * A277886 A359588 A337868

Adjacent sequences: A097243 A097244 A097245 * A097247 A097248 A097249

Keyword

nonn,mult

Author

Reinhard Zumkeller, Aug 03 2004

Status

approved