cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A071786 In prime factorization of n replace each prime with its reversal (in decimal notation).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 31, 14, 15, 16, 71, 18, 91, 20, 21, 22, 32, 24, 25, 62, 27, 28, 92, 30, 13, 32, 33, 142, 35, 36, 73, 182, 93, 40, 14, 42, 34, 44, 45, 64, 74, 48, 49, 50, 213, 124, 35, 54, 55, 56, 273, 184, 95, 60, 16, 26, 63, 64, 155, 66, 76, 284, 96, 70, 17, 72
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 06 2002

Keywords

Comments

The range of A007500 is a subset of the range of this sequence. - Reinhard Zumkeller, Jul 06 2009
Prime factors counted with multiplicity. - Harvey P. Dale, Jul 08 2017

Examples

			a(143) = a(11*13) = a(11)*a(13) = 11*31 = 341.

Links

Crossrefs

Cf. A151764, A161594, A151765. For records see A151766, A151767.
Cf. A151768 (complement), A376858 (fixed points).
Cf. A027746.

Programs

  • Haskell
    a071786 = product . map a004086 . a027746_row
-- Reinhard Zumkeller, Oct 14 2011
  • 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: # R. J. Mathar, Jun 16 2009
# second Maple program:
r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):
a:= n-> mul(r(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 19 2017
  • Mathematica
    Table[Times@@IntegerReverse/@Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]],{n,80}] (* Harvey P. Dale, Jul 08 2017 *)
  • PARI
    rev(n)=fromdigits(Vecrev(digits(n)))
a(n)=my(f=factor(n)); prod(i=1,#f~,rev(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015
  • Python
    from sympy import factorint
from operator import mul
from functools import reduce
def A071786(n):
    return 1 if n==1 else reduce(mul,(int(str(p)[::-1])**e for p,e in factorint(n).items())) # Chai Wah Wu, Aug 14 2014

Formula

Completely multiplicative with a(p) = A004086(p), p prime.
a(A000040(n)) = A004087(n).

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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.

Examples

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

Links

Crossrefs

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

Programs

  • Haskell
    a161594 = a004086 . a071786  -- Reinhard Zumkeller, Oct 14 2011
  • 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 *)
  • 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
  • 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

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)

Extensions

Simpler definition from R. J. Mathar, Jun 16 2009
Edited by N. J. A. Sloane, Jun 23 2009
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