A161594 - OEIS

%I #38 Mar 28 2022 15:35:22

%S 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 72,82,29,3,31,23,33,241,53,63,37,281,39,4,41,24,43,44,54,46,47,84,94,

%U 5,312,421,53,45,55,65,372,481,59,6,61,62,36,46,551,66,67,482,69,7,71,27

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

%C Might be called TITO(n), turning n inside out then turning outside in.

%C 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 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 T. Khovanova, <a href="http://blog.tanyakhovanova.com/?p=144">Turning Numbers Inside Out</a> [From _Tanya Khovanova_, Jul 07 2009]

%F a(p) = p, for prime p.

%F a(A161598(n)) <> A161598(n); a(A161597(n)) = A161597(n); A010051(a(A161600(n))) = 1.

%F From _M. F. Hasler_, Jun 25 2009: (Start)

%F a( p*10^k ) = p for any prime p.

%F 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 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 a(34) = 241, because 34 = 2*17, f(34) = 2*71 = 142, and reversing gives 241.

%p 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 # second Maple program:

%p r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):

%p a:= n-> r(mul(r(i[1])^i[2], i=ifactors(n)[2])):

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 19 2017

%t reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Table[f[n], {n, 100}]

%t 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 (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 (Haskell) a161594 = a004086 . a071786  -- _Reinhard Zumkeller_, Oct 14 2011

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def f(n): return prod(int(str(p)[::-1])**e for p, e in factorint(n).items())

%o def R(n): return int(str(n)[::-1])

%o def a(n): return 1 if n == 1 else R(f(n))

%o print([a(n) for n in range(1, 73)]) # _Michael S. Branicky_, Mar 28 2022

%Y Cf. A161597, A161598, A161600, A071786, A004086, A151764.

%K nonn,base,nice,look

%O 1,2

%A _J. H. Conway_ & _Tanya Khovanova_, Jun 14 2009

%E Simpler definition from _R. J. Mathar_, Jun 16 2009

%E Edited by _N. J. A. Sloane_, Jun 23 2009