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(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)
(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))))))
;; 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
KEYWORD
nonn,mult
AUTHOR
Reinhard Zumkeller, Aug 03 2004
STATUS
approved