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

%I A124859 #36 Feb 16 2025 08:33:03
%S A124859 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,
%T A124859 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,
%U A124859 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
%N A124859 Multiplicative with p^e -> primorial(e), p prime and e > 0.
%H A124859 Indranil Ghosh, <a href="/A124859/b124859.txt">Table of n, a(n) for n = 1..5000</a> (first 1000 terms from R. Zumkeller)
%H A124859 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactorization.html">Prime Factorization</a>
%H A124859 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>
%H A124859 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A124859 a(A000040(x)^n) = A002110(n); a(A002110(n)) = A000079(n);
%F A124859 a(A005117(n)) = 2^A001221(A005117(n)) = A072048(n);
%F A124859 A001221(a(n)) = A051903(n); A001222(a(n)) = A001222(n).
%F A124859 From _Antti Karttunen_, Mar 06 2017: (Start)
%F A124859 a(1) = 1, for n > 1, a(n) = A002110(A067029(n)) * a(A028234(n)).
%F A124859 a(n) = A278159(A156552(n)).
%F A124859 a(A278159(n)) = A278222(n).
%F A124859 a(a(n)) = A046523(n). [after _Matthew Vandermast_'s May 19 2012 formula for the latter sequence]
%F A124859 A181819(a(n)) = A238745(n). [after _Matthew Vandermast_'s formula for the latter sequence]
%F A124859 (End)
%F A124859 a(n) = A108951(A181819(n)). [Primorial inflation of the prime shadow of n] - _Antti Karttunen_, Sep 15 2023
%e A124859 From _Michael De Vlieger_, Mar 06 2017: (Start)
%e A124859 a(2) = 2 since 2 = 2^1, thus primorial p_1# = 2.
%e A124859 a(4) = 6 since 4 = 2^2, thus primorial p_2# = 2*3 = 6.
%e A124859 a(6) = 4 because 6 is squarefree with omega(6)=2, thus 2^2 = 4.
%e A124859 a(8) = 30 since 8 = 2^3, thus primorial p_3# = 2*3*5 = 30.
%e A124859 a(10) = 4 since 10 is squarefree with omega(10)=2, thus 2^2 = 4.
%e A124859 a(12) = 12 since 12 = 2^1 * 3^2, thus primorials p_1# * p_2# = 2*6 = 12.
%e A124859 (End)
%p A124859 A124859 := proc(n)
%p A124859     local a,pf;
%p A124859     a := 1;
%p A124859     for pf in ifactors(n)[2] do
%p A124859         a := a*A002110(pf[2]) ;
%p A124859     end do:
%p A124859     a ;
%p A124859 end proc:
%p A124859 seq(A124859(n),n=1..80) ; # _R. J. Mathar_, Oct 06 2017
%t A124859 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 *)
%o A124859 (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
%o A124859 (Scheme) (define (A124859 n) (cond ((= 1 n) 1) (else (* (A002110 (A067029 n)) (A124859 (A028234 n)))))) ;; _Antti Karttunen_, Mar 06 2017
%o A124859 (Python)
%o A124859 from sympy.ntheory.factor_ import core
%o A124859 from sympy import factorint, primorial, primefactors
%o A124859 from operator import mul
%o A124859 def omega(n): return 0 if n==1 else len(primefactors(n))
%o A124859 def a(n):
%o A124859     f=factorint(n)
%o A124859     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
%Y A124859 Cf. A000040, A000079, A001221, A001222, A002110, A005117, A028234, A046523, A067029, A072048, A108951, A156552, A181819, A181821, A238745, A278159, A278222.
%K A124859 nonn,mult
%O A124859 1,2
%A A124859 _Reinhard Zumkeller_, Nov 10 2006