A161600 -id:A161600 - cp's OEIS frontend

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

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

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.

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

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]

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

a161594 = a004086 . a071786  -- Reinhard Zumkeller, Oct 14 2011

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

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 *)

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

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

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)

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

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

A161597 Numbers such that TITO(n) = n, where TITO(n) = A161594(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 37, 39, 41, 43, 44, 46, 47, 53, 55, 59, 61, 62, 66, 67, 69, 71, 73, 77, 79, 82, 83, 86, 88, 89, 93, 97, 99, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

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

TITO(p) = p, for any prime p.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Complement of A161598; nonprimes: A161600.

a161597 n = a161597_list !! (n-1)
a161597_list = filter (\x -> a161594 x == x) [1..]
-- Reinhard Zumkeller, Oct 14 2011

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] Select[Range[200], f[ # ] == # &]

is(n)={n==A161594(n)} \\ M. F. Hasler, May 11 2015

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

Offset corrected by Reinhard Zumkeller, Oct 14 2011

A161730 Palindromic numbers that are fixed points of the TITO operation (see A161594) and are not products of palindromic primes.

Original entry on oeis.org

72927, 76167, 434434, 868868, 1226221, 4778774, 5703075, 8755578, 9386839, 13488431, 43877834, 123848321, 564414465, 777555777, 1072772701, 1946776491, 9935115399, 12467976421, 52854045825, 74663436647, 83361616338, 95829592859

Offset: 1

Tanya Khovanova, Jun 17 2009

The numbers in this sequence are palindromic numbers that are fixed points of the TITO operation and are not primes and are not in A046351.

M. F. Hasler, Table of n, a(n) for n = 1..35. [From _M. F. Hasler_, Jun 25 2009]
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Cf. A161594, A161597, A161598, A161600.

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] rev[n_] := FromDigits[Reverse[IntegerDigits[n]]] Select[Range[5000000], rev[ # ] == # && ! PrimeQ[ # ] && f[ # ] == # && Map[rev, Transpose[FactorInteger[ # ]][[1]]] != Transpose[FactorInteger[ # ]][[1]] &]

for( d=1,19, my(p=10^((d+1)\2),q=10^(d%2)); for( i=p\10,p-1, my(n = i\q*p+R(i),f); A161594(n)==n || next; apply(R,f=factor(n)[,1])==f && next; print1(n",") )) /* uses definitions given in A161594 */ \\ M. F. Hasler, Jun 25 2009

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

Terms beyond a(6) from M. F. Hasler, Jun 25 2009

A161721 Primes p such that the reversal of p is prime and the product of p with its reversal is a palindrome.

Original entry on oeis.org

2, 3, 11, 101, 1021, 1201, 111211, 112111, 1000211, 1010201, 1020101, 1101211, 1102111, 1111021, 1112011, 1120001, 1121011, 1201111, 10011101, 10012001, 10021001, 10100201, 10111001, 10200101, 11012011, 11021011, 11100121, 12100111

Offset: 1

Tanya Khovanova, Jun 17 2009

This sequence is a subsequence of A062936. If you multiply a member of this sequence by its reversal you get a number fixed under TITO algorithm (see A161594).
Conjecture: except for a(2) which equals 3, all terms can only be composed of the digits 0, 1 or 2. - Chai Wah Wu, Jan 07 2015
Conjecture: the digit 2 can only appear once in each term. - Robert G. Wilson v, Jan 07 2015
Number of terms less than 10^n: 2, 3, 4, 6, 6, 8, 18, 28, 37, 65, 97, 153, 230, 304, 414, 556, 756, 960, 1255, ... - Robert G. Wilson v, Jan 07 2015
A proper subset of A007500. - Robert G. Wilson v, Jan 07 2015

			1021 is a prime number, its reversal is 1201, which is also a prime. The product 1021*1201 = 1226221 is a palindrome.

Robert G. Wilson v, Table of n, a(n) for n = 1..1255 (first 97 terms from Chai Wah Wu)
Tanya Khovanova, Turning Numbers Inside Out

Cf. A161594, A161597, A161600.

rev := proc (n) local nn: nn := convert(n, base, 10): add(nn[j]*10^(nops(nn)-j), j = 1 .. nops(nn)) end proc: a := proc (n) local p: p := ithprime(n): if isprime(rev(p)) = true and rev(p*rev(p)) = p*rev(p) then p else end if end proc: seq(a(n), n = 1 .. 800000); # Emeric Deutsch, Jun 26 2009

rev[n_]:=FromDigits[Reverse[IntegerDigits[n]]]; t={}; Do[p=Prime[n]; If[PrimeQ[q=rev[p]] && rev[p*q]==p*q, AppendTo[t,p]], {n,8*10^5}]; t (* Jayanta Basu, May 11 2013 *)

from sympy import isprime
A161721_list = [2]
for i in range(3,10**6,2):
    j = int(str(i)[::-1])
    if j == i:
        s = str(i**2)
        if s == s[::-1] and isprime(i):
            A161721_list.append(i)
    elif j > i:
        s = str(i*j)
        if s == s[::-1] and isprime(i) and isprime(j):
            A161721_list.extend([i,j])
A161721_list = sorted(A161721_list) # Chai Wah Wu, Jan 07 2015

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

More terms from Emeric Deutsch, Jun 26 2009

A161732 Fixed points of the TITO operation (A161594) that are also composite palindromes.

Original entry on oeis.org

4, 6, 8, 9, 22, 33, 44, 55, 66, 77, 88, 99, 121, 202, 242, 252, 262, 303, 343, 363, 393, 404, 484, 505, 525, 606, 616, 626, 686, 707, 808, 909, 939, 1111, 1331, 1441, 1661, 1991, 2112, 2222, 2662, 2772, 2882, 3333, 3443, 3773, 3883, 3993, 4224, 4444, 5445

Offset: 1

Tanya Khovanova, Jun 17 2009

This sequence is a proper superset of A046351 (palindromic composite numbers with only palindromic prime factors). The smallest number that doesn't belong to A046351 is 72927. The numbers that are in this sequence and are not in A046351 are given in A161730.

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

Cf. A161594, A161597, A161598, A161600, A046351

reversepower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n]]]^k f[n_] := FromDigits[ Reverse[IntegerDigits[Times @@ Map[reversepower, FactorInteger[n]]]]] rev[n_] := FromDigits[Reverse[IntegerDigits[n]]] Select[Range[10000], f[ # ] == # && rev[ # ] == # && ! PrimeQ[ # ] &]

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

A257842 Semiprimes pq such that R(pq) = R(p)*R(q), where R = A004086 = reverse digits.

Original entry on oeis.org

4, 6, 9, 22, 26, 33, 39, 46, 55, 62, 69, 77, 82, 86, 93, 121, 143, 169, 187, 202, 206, 226, 253, 262, 299, 303, 309, 339, 341, 393, 422, 446, 451, 466, 473, 482, 505, 583, 622, 626, 633, 662, 669, 671, 699, 707, 781, 802, 842, 862, 866, 886, 933, 939, 961

Offset: 1

M. F. Hasler, May 11 2015

A subsequence of A161600. Almost all terms with less than 4 digits are either multiples of 2 or 3 or of 11.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A161600, A071786, A004086.

N:= 1000: # to get all terms <= N
digrev:= proc(n) local L,i;
L:= convert(n,base,10);
add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
F:= proc(p,q) if digrev(p*q)=digrev(p)*digrev(q) then p*q else NULL fi end proc:
sort([seq(seq(F(Primes[i],q), q = select(`<=`,Primes[i..-1],N/Primes[i])), i=1..nops(Primes))]); # Robert Israel, May 14 2015

f[n_]:=FactorInteger[n][[1,1]];g[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Select[Range@1000,PrimeOmega[#]==2&&g[f[#]*#/f[#]]==g[f[#]]*g[#/f[#]]&] (* Ivan N. Ianakiev, May 14 2015 *)

is(n)=bigomega(n)==2&&!eval(concat(Vecrev(Str(n"-"vecmin(n=factor(n)[,1])"*"vecmax(n)))))

A162151 Numbers n such that m=TITO(n)>n and TITO(m)=n, where TITO() = A161594().

Original entry on oeis.org

12, 18, 24, 27, 36, 45, 48, 132, 144, 156, 198, 264, 276, 288, 291, 297, 372, 375, 396, 405, 492, 495, 528, 576, 1089, 1212, 1236, 1287, 1356, 1359, 1452, 1572, 1584, 1629, 1683, 1728, 1812, 1818, 2002, 2067, 2079, 2178, 2304, 2424, 2532, 2676, 2721, 2727

Offset: 1

Zak Seidov, Jun 26 2009

Or, numbers that end in two-cycles under TITO operation.

			For smaller n's, m is a reversal of n, but for larger n's, there are other cases as well:{12,21},{18,81},{24,42},{27,72},...,{291,732},...,{372,651}, etc.

Zak Seidov, 141 pairs {n,m} for n<30000
T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Cf. A161594, A161597, A161598, A161600, A161730, A161732.

m=A161594(n)>n, and A161594(m)=n.

cp's OEIS Frontend

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

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A161597 Numbers such that TITO(n) = n, where TITO(n) = A161594(n).

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A161730 Palindromic numbers that are fixed points of the TITO operation (see A161594) and are not products of palindromic primes.

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A161721 Primes p such that the reversal of p is prime and the product of p with its reversal is a palindrome.

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A161732 Fixed points of the TITO operation (A161594) that are also composite palindromes.

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A257842 Semiprimes p*q such that R(p*q) = R(p)*R(q), where R = A004086 = reverse digits.

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A162151 Numbers n such that m=TITO(n)>n and TITO(m)=n, where TITO() = A161594().

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A257842 Semiprimes pq such that R(pq) = R(p)*R(q), where R = A004086 = reverse digits.