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

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, 72, 82, 29, 3, 31, 23, 33, 241, 53, 63, 37, 281, 39, 4, 41, 24, 43, 44, 54, 46, 47, 84, 94, 5, 312, 421, 53, 45, 55, 65, 372, 481, 59, 6, 61, 62, 36, 46, 551, 66, 67, 482, 69, 7, 71, 27

Offset

1,2

Comments

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

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.

Links

M. F. Hasler, Table of n, a(n) for n=1..5000. [From M. F. Hasler, Jun 24 2009]

T. Khovanova, Turning Numbers Inside Out [From Tanya Khovanova, Jul 07 2009]

Formula

a(p) = p, for prime p.

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

From M. F. Hasler, Jun 25 2009: (Start)

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

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;

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)

Example

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

Maple

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
# second Maple program:
r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):
a:= n-> r(mul(r(i[1])^i[2], i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 19 2017

Mathematica

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Table[f[n], {n, 100}]
Table[IntegerReverse[Times@@Flatten[Table[IntegerReverse[#[[1]]], #[[2]]]& /@FactorInteger[n]]], {n, 100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 21 2016 *)

Prog

(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
(Haskell) a161594 = a004086 . a071786  -- Reinhard Zumkeller, Oct 14 2011
(Python)
from math import prod
from sympy import factorint
def f(n): return prod(int(str(p)[::-1])**e for p, e in factorint(n).items())
def R(n): return int(str(n)[::-1])
def a(n): return 1 if n == 1 else R(f(n))
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Mar 28 2022

Crossrefs

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

Sequence in context: A151765 A343750 A107603 * A084011 A345110 A004086

Adjacent sequences: A161591 A161592 A161593 * A161595 A161596 A161597

Keyword

nonn,base,nice,look

Author

J. H. Conway & Tanya Khovanova, Jun 14 2009

Extensions

Simpler definition from R. J. Mathar, Jun 16 2009

Edited by N. J. A. Sloane, Jun 23 2009

Status

approved