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A161594 a(n) = R(f(n)), where R = A004086 = reverse (decimal) digits, f = A071786 = reverse digits of prime factors.

%I A161594 #38 Mar 28 2022 15:35:22
%S A161594 1,2,3,4,5,6,7,8,9,1,11,21,13,41,51,61,17,81,19,2,12,22,23,42,52,26,
%T A161594 72,82,29,3,31,23,33,241,53,63,37,281,39,4,41,24,43,44,54,46,47,84,94,
%U A161594 5,312,421,53,45,55,65,372,481,59,6,61,62,36,46,551,66,67,482,69,7,71,27
%N A161594 a(n) = R(f(n)), where R = A004086 = reverse (decimal) digits, f = A071786 = reverse digits of prime factors.
%C A161594 Might be called TITO(n), turning n inside out then turning outside in.
%C A161594 Here is the operation: take a number n and find its prime factors. Reverse the digits of every prime factor (for example, replace 17 by 71). Multiply the factors respecting multiplicities. For example, if the original number was 17^2*43^3, the new product will be 71^2*34^3. After that, reverse the resulting number.
%H A161594 M. F. Hasler, <a href="/A161594/b161594.txt">Table of n, a(n) for n=1..5000</a>. [From _M. F. Hasler_, Jun 24 2009]
%H A161594 T. Khovanova, <a href="http://blog.tanyakhovanova.com/?p=144">Turning Numbers Inside Out</a> [From _Tanya Khovanova_, Jul 07 2009]
%F A161594 a(p) = p, for prime p.
%F A161594 a(A161598(n)) <> A161598(n); a(A161597(n)) = A161597(n); A010051(a(A161600(n))) = 1.
%F A161594 From _M. F. Hasler_, Jun 25 2009: (Start)
%F A161594 a( p*10^k ) = p for any prime p.
%F A161594 Proof: if gcd( p, 2*5) = 1, then a( p * 10^k ) = R( R(p) * R(2)^k * R(5)^k ) = R( R(p) * 10^k ) = R(R(p)) = p;
%F A161594 if gcd(p, 2*5) = 2, then p=2 and a( p * 10^k ) = R( R(2)^(k+1) * R(5)^k ) = R( 2 * 10^k ) = 2 = p and mutatis mutandis for gcd(p, 2*5) = 5. (End)
%e A161594 a(34) = 241, because 34 = 2*17, f(34) = 2*71 = 142, and reversing gives 241.
%p A161594 read("transforms") ; A071786 := proc(n) local ifs,a,d ; ifs := ifactors(n)[2] ; a := 1 ; for d in ifs do a := a*digrev(op(1,d))^op(2,d) ; od: a ; end: A161594 := proc(n) digrev(A071786(n)) ; end: seq(A161594(n),n=1..80) ; # _R. J. Mathar_, Jun 16 2009
%p A161594 # second Maple program:
%p A161594 r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):
%p A161594 a:= n-> r(mul(r(i[1])^i[2], i=ifactors(n)[2])):
%p A161594 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 19 2017
%t A161594 reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Table[f[n], {n, 100}]
%t A161594 Table[IntegerReverse[Times@@Flatten[Table[IntegerReverse[#[[1]]],#[[2]]]& /@FactorInteger[n]]],{n,100}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 21 2016 *)
%o A161594 (PARI) R=A004086; A161594(n)={n=factor(n);n[,1]=apply(R,n[,1]);R(factorback(n))} \\  _M. F. Hasler_, Jun 24 2009. Removed code for R here, see A004086 for most recent & efficient version. - _M. F. Hasler_, May 11 2015
%o A161594 (Haskell) a161594 = a004086 . a071786  -- _Reinhard Zumkeller_, Oct 14 2011
%o A161594 (Python)
%o A161594 from math import prod
%o A161594 from sympy import factorint
%o A161594 def f(n): return prod(int(str(p)[::-1])**e for p, e in factorint(n).items())
%o A161594 def R(n): return int(str(n)[::-1])
%o A161594 def a(n): return 1 if n == 1 else R(f(n))
%o A161594 print([a(n) for n in range(1, 73)]) # _Michael S. Branicky_, Mar 28 2022
%Y A161594 Cf. A161597, A161598, A161600, A071786, A004086, A151764.
%K A161594 nonn,base,nice,look
%O A161594 1,2
%A A161594 _J. H. Conway_ & _Tanya Khovanova_, Jun 14 2009
%E A161594 Simpler definition from _R. J. Mathar_, Jun 16 2009
%E A161594 Edited by _N. J. A. Sloane_, Jun 23 2009