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A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

%I A071364 #28 Sep 17 2022 02:00:29
%S A071364 1,2,2,4,2,6,2,8,4,6,2,12,2,6,6,16,2,18,2,12,6,6,2,24,4,6,8,12,2,30,2,
%T A071364 32,6,6,6,36,2,6,6,24,2,30,2,12,12,6,2,48,4,18,6,12,2,54,6,24,6,6,2,
%U A071364 60,2,6,12,64,6,30,2,12,6,30,2,72,2,6,18,12,6,30,2,48,16,6,2,60,6,6,6,24
%N A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.
%C A071364 A046523(a(n))=A046523(n); A046523(n)<=a(n)<=n; A001221(a(n))=A001221(n), A001222(a(n))=A001222(n); A020639(a(n))=2, A006530(a(n))=A000040(A001221(n))<=A006530(n); A000005(a(n))=A000005(n);
%C A071364 a(a(n))=a(n); a(n)=2^k iff n=p^k, p prime, k>0 (A000961); if n>1 is not a prime power, then a(n) mod 6 = 0; range of values = A055932, as distinct prime factors of a(n) are consecutive: a(n)=n iff n=A055932(k) for some k;
%C A071364 a(A003586(n))=A003586(n).
%H A071364 Daniel Forgues, <a href="/A071364/b071364.txt">Table of n, a(n) for n=1..100000</a>
%F A071364 In prime factorization of n, replace least prime by 2, next least by 3, etc.
%F A071364 a(n) = product(A000040(k)^A124010(k): k=1..A001221(n)). - _Reinhard Zumkeller_, Apr 27 2013
%e A071364 a(105875) = a(5*5*5*7*11*11) = 2*2*2*3*5*5 = 600.
%t A071364 Table[ e = Last /@ FactorInteger[n]; Product[Prime[i]^e[[i]], {i, Length[e]}], {n, 88}] (* _Ray Chandler_, Sep 23 2005 *)
%o A071364 (Haskell)
%o A071364 a071364 = product . zipWith (^) a000040_list . a124010_row
%o A071364 -- _Reinhard Zumkeller_, Feb 19 2012
%o A071364 (PARI) a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ _Michel Marcus_, Jun 13 2014
%o A071364 (Python)
%o A071364 from math import prod
%o A071364 from sympy import prime, factorint
%o A071364 def A071364(n): return prod(prime(i+1)**p[1] for i,p in enumerate(sorted(factorint(n).items()))) # _Chai Wah Wu_, Sep 16 2022
%Y A071364 Cf. A071365, A071366.
%Y A071364 Cf. A000040.
%Y A071364 Cf. A085079, A089247.
%Y A071364 The range is A055932.
%Y A071364 The reversed version is A331580.
%Y A071364 Unsorted prime signature is A124010.
%Y A071364 Numbers whose prime signature is aperiodic are A329139.
%Y A071364 Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.
%K A071364 nonn
%O A071364 1,2
%A A071364 _Reinhard Zumkeller_, May 21 2002
%E A071364 Extended by _Ray Chandler_, Sep 23 2005