A124859 Multiplicative with p^e -> primorial(e), p prime and e > 0.

1, 2, 2, 6, 2, 4, 2, 30, 6, 4, 2, 12, 2, 4, 4, 210, 2, 12, 2, 12, 4, 4, 2, 60, 6, 4, 30, 12, 2, 8, 2, 2310, 4, 4, 4, 36, 2, 4, 4, 60, 2, 8, 2, 12, 12, 4, 2, 420, 6, 12, 4, 12, 2, 60, 4, 60, 4, 4, 2, 24, 2, 4, 12, 30030, 4, 8, 2, 12, 4, 8, 2, 180, 2, 4, 12, 12, 4, 8, 2, 420, 210, 4, 2, 24, 4, 4

Offset: 1

Author

Reinhard Zumkeller, Nov 10 2006

Keywords

• nonn
• mult

Examples

			From _Michael De Vlieger_, Mar 06 2017: (Start)
a(2) = 2 since 2 = 2^1, thus primorial p_1# = 2.
a(4) = 6 since 4 = 2^2, thus primorial p_2# = 2*3 = 6.
a(6) = 4 because 6 is squarefree with omega(6)=2, thus 2^2 = 4.
a(8) = 30 since 8 = 2^3, thus primorial p_3# = 2*3*5 = 30.
a(10) = 4 since 10 is squarefree with omega(10)=2, thus 2^2 = 4.
a(12) = 12 since 12 = 2^1 * 3^2, thus primorials p_1# * p_2# = 2*6 = 12.
(End)
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..5000 (first 1000 terms from R. Zumkeller)
• Eric Weisstein's World of Mathematics, Prime Factorization
• Eric Weisstein's World of Mathematics, Primorial
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000040, A000079, A001221, A001222, A002110, A005117, A028234, A046523, A067029, A072048, A108951, A156552, A181819, A181821, A238745, A278159, A278222.

Programs

Maple

A124859 := proc(n)
    local a,pf;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*A002110(pf[2]) ;
    end do:
    a ;
end proc:
seq(A124859(n),n=1..80) ; # R. J. Mathar, Oct 06 2017

Mathematica

Table[Which[n == 1, 1, SquareFreeQ@ n, 2^PrimeNu@ n, True, Times @@ Map[Times @@ Prime@ Range@ # &, #[[All, -1]]]] &@ FactorInteger@ n, {n, 86}] (* Michael De Vlieger, Mar 06 2017 *)

PARI

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k,1] = prod(j=1, f[k,2], prime(j)); f[k,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 16 2015

Python

from sympy.ntheory.factor_ import core
from sympy import factorint, primorial, primefactors
from operator import mul
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    f=factorint(n)
    return n if n<3 else 2**omega(n) if core(n) == n else reduce(mul, [primorial(f[i]) for i in f]) # Indranil Ghosh, May 13 2017

Scheme

(define (A124859 n) (cond ((= 1 n) 1) (else (* (A002110 (A067029 n)) (A124859 (A028234 n)))))) ;; Antti Karttunen, Mar 06 2017

Formula

a(A000040(x)^n) = A002110(n); a(A002110(n)) = A000079(n);

a(A005117(n)) = 2^A001221(A005117(n)) = A072048(n);

A001221(a(n)) = A051903(n); A001222(a(n)) = A001222(n).

From Antti Karttunen, Mar 06 2017: (Start)

a(1) = 1, for n > 1, a(n) = A002110(A067029(n)) * a(A028234(n)).

a(n) = A278159(A156552(n)).

a(A278159(n)) = A278222(n).

a(a(n)) = A046523(n). [after Matthew Vandermast's May 19 2012 formula for the latter sequence]

A181819(a(n)) = A238745(n). [after Matthew Vandermast's formula for the latter sequence]

(End)

a(n) = A108951(A181819(n)). [Primorial inflation of the prime shadow of n] - Antti Karttunen, Sep 15 2023