A161594 -id:A161594 - cp's OEIS frontend

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

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

A161600 Nonprime numbers such that TITO(n) = n, where TITO(n) = A161594(n).

Original entry on oeis.org

1, 4, 6, 8, 9, 22, 26, 33, 39, 44, 46, 55, 62, 66, 69, 77, 82, 86, 88, 93, 99, 121, 143, 169, 187, 202, 206, 226, 242, 252, 253, 262, 286, 299, 303, 309, 339, 341, 343, 363, 393, 404, 422, 446, 451, 466, 473, 482, 484, 505, 525, 583, 606, 616, 622, 626, 633, 662, 669, 671, 682, 686

Offset: 1

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

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

Cf. A010051, A161594; subsequence of A161597.

a161600 n = a161600_list !! (n-1)
a161600_list = filter ((== 0) . a010051) a161597_list
-- 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[600], f[#] == # && ! PrimeQ[#] &]

is(n)=!isprime(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
Minor edits and more displayed terms from M. F. Hasler, May 11 2015

A161598 Numbers such that TITO(n) is not equal to n, where TITO(n) = A161594(n).

Original entry on oeis.org

10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 34, 35, 36, 38, 40, 42, 45, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 63, 64, 65, 68, 70, 72, 74, 75, 76, 78, 80, 81, 84, 85, 87, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116

Offset: 1

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

There are no prime numbers in the sequence: A010051(a(n)) = 0.

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 A161597.

a161598 n = a161598_list !! (n-1)
a161598_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[ # ] != # &]

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

A161955 TITO2(n): The operation A161594 in binary, digit-reversals carried out in base 2.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 19, 13, 27, 7, 29, 15, 31, 1, 57, 17, 49, 9, 37, 19, 33, 5, 41, 21, 43, 11, 45, 23, 47, 3, 35, 19, 51, 13, 53, 27, 65, 7, 105, 29, 59, 15, 61, 31, 63, 1, 59, 57, 67, 17, 117, 49, 71, 9, 73, 37, 105, 19, 109

Offset: 1

Tanya Khovanova, Jun 22 2009

The TITO function in binary: Represent n as a product of its prime factors in binary.

Revert the binary digits of each of these factors, then multiply them with the same multiplicities as in n--so the base-2 representation does not affect the exponents in the canonical prime factorization. Reverse the product in binary to get a(n).

			To calculate TITO2(n=99): 99 = 3^3*11. Prime factors 3 and 11 in binary are 11 and 1011 correspondingly. Reversing those numbers we get 11 and 1101. The product with multiplicities is the binary product of 11*11*1101 = 1110101. Reversing that we get 1010111, which corresponds to 87. Hence a(99) = 87.

Alois P. Heinz, Table of n, a(n) for n = 1..40000
Tanya Khovanova, Turning Numbers Inside Out

Cf. A161594.

r:= proc(n) local m, t; m, t:=n, 0; while m>0
      do t:=2*t+irem(m, 2, 'm') od; t end:
a:= n-> r(mul(r(i[1])^i[2], i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 29 2017

reverseBinPower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n, 2]], 2]^k fBin[n_] := FromDigits[ Reverse[IntegerDigits[ Times @@ Map[reverseBinPower, FactorInteger[n]], 2]], 2] Table[fBin[n], {n, 200}]

a(n) = A030101(A162742(n)) - R. J. Mathar, Aug 03 2009

Edited by R. J. Mathar, Aug 03 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

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.

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

Reinhard Zumkeller, Jun 06 2002

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

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

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

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

a071786 = product . map a004086 . a027746_row
-- 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: # 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

Table[Times@@IntegerReverse/@Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]],{n,80}] (* Harvey P. Dale, Jul 08 2017 *)

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

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

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

a(A000040(n)) = A004087(n).

A151765 a(n) = f(R(n)), where f(n) = A071786(n), R(n) = A004086(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 13, 14, 213, 16, 17, 81, 217, 2, 12, 22, 32, 42, 124, 26, 72, 28, 128, 3, 31, 32, 33, 34, 35, 63, 37, 38, 39, 4, 14, 24, 142, 44, 54, 64, 146, 84, 148, 5, 15, 25, 35, 45, 55, 155, 75, 355, 455, 6, 16, 62, 36, 64, 56, 66, 364, 68, 96, 7, 71, 27

Offset: 1

N. J. A. Sloane, Jun 22 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A071786, A004086, A161594.

a151765 = a071786 . a004086 -- Reinhard Zumkeller, Oct 14 2011

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

cp's OEIS Frontend

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

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A161600 Nonprime numbers such that TITO(n) = n, where TITO(n) = A161594(n).

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A161598 Numbers such that TITO(n) is not equal to 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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A161955 TITO2(n): The operation A161594 in binary, digit-reversals carried out in base 2.

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

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

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

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