A004086 -id:A004086 - cp's OEIS frontend

A077741 Smallest multiple of n which begins with R(n) and ends in n where R(n) (A004086) is the digit reversal of n. Suitable number of zeros are assumed to the left of the MSD if required.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2112, 31213, 41314, 5115, 61216, 71417, 8118, 91219, 20, 12621, 22, 32223, 4224, 525, 62426, 72927, 82628, 92829, 30, 130231, 23232, 33, 430134, 53235, 6336, 732637, 83638, 93639, 40, 143541, 241542, 34443, 44, 5445

Offset: 1

Amarnath Murthy, Nov 20 2002

			a(14) = 41314 = 14*2951, a(10) = 10 as it can be written as 010.

Cf. A077742.

Corrected and extended by Ray Chandler, Jun 11 2003

A100414 Numbers n such that n is R(n)-th composite number where R(n) is the digit reversal of n (A002808(A004086(n))=n).

Original entry on oeis.org

21, 48034, 69926, 180461, 214591, 409473, 563715, 41630193, 253385633342, 661494322636

Offset: 1

Farideh Firoozbakht, Dec 10 2004

There is no further term < 3*10^9.

a(11) > 3*10^12. [Donovan Johnson, Jun 14 2009]

			41630193 is in the sequence because 41630193 is the 39103614th composite number.

Cf. A002808, A004086, A100412, A100413, A100415.

Do[s=FromDigits[Reverse[IntegerDigits[n]]];If[s

a(9)-a(10) from Donovan Johnson, Jun 14 2009

A100415 Numbers n such that n is R(n)-th nonprime number, where R(n) is the digit reversal of n (A018252(A004086(n))=n).

Original entry on oeis.org

1, 64, 524, 534, 58725, 907538, 6264385, 9438088, 9596598, 27895162, 422984004, 548911025, 8804661048, 49640253574, 63899981216, 95138721219, 97895906839, 469449672154

Offset: 1

Farideh Firoozbakht, Dec 12 2004

n is in the sequence iff n is not prime and R(n)=n-pi(n). There is no further term up to 3670000000.

a(19) > 10^13. Up to that limit, this sequence contains all the numbers k such that R(k) = k - pi(k). - Giovanni Resta, Aug 08 2019

			548911025 is in the sequence because 548911025 is the 520118945th nonprime natural number.

Cf. A018252, A004086, A100413, A100414, A114926.

Do[s = FromDigits[Reverse[IntegerDigits[n]]]; If[ ! PrimeQ[n] && s == n - PrimePi[n], Print[n]], {n, 548911025}]

a(13)-a(18) from Giovanni Resta, Aug 08 2019

A104154 For each natural number n: if the last digit of n is not zero and A004086(n) is prime, append A004086(n) to the sequence.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 41, 61, 71, 13, 23, 43, 53, 73, 83, 17, 37, 47, 67, 97, 19, 29, 59, 79, 89, 101, 401, 601, 701, 211, 311, 811, 911, 421, 521, 821, 131, 331, 431, 631, 241, 541, 641, 941, 151, 251, 751, 461, 661, 761, 271, 571, 971, 181, 281, 881

Offset: 1

Cino Hilliard, Mar 09 2005

Equivalently, these are the prime numbers ordered by their reversal. - Rémy Sigrist, Feb 13 2022

			The last digit of 13 is not '0' and 31 is prime, therefore we append 31.

Rémy Sigrist, Table of n, a(n) for n = 1..9592

Cf. A004087.

a = Select[Range[196], IntegerDigits[ # ][[ -1]] != 0 && PrimeQ[FromDigits[Reverse[ IntegerDigits[ # ]]]] &]; b = {}; For[n = 1, n < Length[a] + 1, n++, AppendTo[b, FromDigits[Reverse[IntegerDigits[a[[n]]]]]]]; b

left(str, n) = { my(v, tmp, x); v =""; tmp = Vec(str); ln=length(tmp); if(n > ln, n=ln); for(x=1, n, v=concat(v, tmp[x]); ); return(v) } \\ Get the left n characters from string str
rev(str) = { local(tmp, s, j); tmp = Vec(Str(str)); s=""; forstep(j=length(tmp), 1, -1, s=concat(s, tmp[j])); return(s) } \\ Get the reverse of the input string
rprime(n) = { local(x, y, v); for(x=1, n, y=rev(x); v=Vec(y); if(left(y, 1)<> "0"&&isprime(eval(y)), print1(y", ")) ) }

from itertools import count, islice
from sympy import primerange
def A104154_gen(): # generator of terms
    yield from (int(d[::-1]) for l in count(1) for d in sorted(str(m)[::-1] for m in primerange(10**(l-1),10**l)))
A104154_list = list(islice(A104154_gen(),20)) # Chai Wah Wu, Feb 17 2022

Edited by Stefan Steinerberger, Aug 01 2007

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

A165696 a(n) is the smallest number m such that the n numbers A004086(m^k) (digit reversal of m^k) for 0 < k < n+1 are all primes.

Original entry on oeis.org

2, 14, 325, 3244, 3244, 110218462, 32149366346, 10212002596432

Offset: 1

Farideh Firoozbakht, Sep 29 2009

			Both numbers A004086(14) = 41 and A004086(14^2) = 691 are primes and 14 is the smallest number with this property so a(2) = 14.

Cf. A004086, A118212, A118213, A165697, A165698, A165699.

a(8) from Matthias Baur, Mar 03 2021

A209914 Number of ways n can be written as a multiple of its reversal A004086(n) +/- a prime p < n.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, 0, 2, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2

Offset: 0

M. F. Hasler, Mar 15 2012

If n is not a multiple of 10, then a(n) < 20. The subsequence { a(10k) }, however, is unbounded. In particular, a(10^k)=2*A006880(k).

			a(10)=8 because R(10)=01=1 and 10 = 3*1 + 7 = 5*1 + 5 = 7*1 + 3 = 8*1 + 2 = 12*1 - 2 = 13*1 - 3 = 15*1 - 5 = 17*1 - 7.

Cf. A209063.

a(n)={my(r=A004086(n)); sum(k=1, (2*n-1)\(r+!r), isprime(abs(n-k*r)))}

A237912 Smallest number m (not ending in a 0) such that m and its digit reversal A004086(m) both have n prime factors (counted with multiplicity).

Original entry on oeis.org

13, 15, 117, 126, 1386, 2576, 21708, 25515, 21168, 46848, 295245, 2937856, 6351048, 21989376, 217340928, 2154281472, 2196652032, 21120051456, 21122906112, 40915058688, 274148425728, 2150086519296, 2707602702336, 6167442456576, 21907217055744, 29798871072768, 420127895977984

Offset: 1

Derek Orr, Feb 15 2014

Palindromes are not included in this sequence since the reverse of a palindrome is the same number. See A076886 and A237913.

			13 and 31 are both prime so a(1) = 13.
15 and 51 have two prime factors (3*5 and 3*17 respectively), so a(2) = 15.

Cf. A004086, A076886, A237913.

import sympy
from sympy import factorint
def rev(x):
  rev = ''
  for i in str(x):
    rev = i + rev
  return int(rev)
def RevFact(x):
  n = 1
  while n < 10**8:
    if rev(n) != n:
      if n % 10 != 0:
        if sum(list(factorint(n).values())) == x:
          if sum(list(factorint(rev(n)).values())) == x:
            return n
          else:
            n += 1
        else:
          n += 1
      else:
        n += 1
    else:
      n += 1
x = 1
while x < 100:
  print(RevFact(x))
  x += 1

a(15)-a(21) from Giovanni Resta, Feb 23 2014

a(22)-a(27) from Max Alekseyev, Feb 07 2024

A248327 Levenshtein distance of n and its reversal in decimal representation, cf. A004086.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 0, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 0, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 0, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 0, 2, 2, 1, 2, 2, 2, 2, 2

Offset: 0

Reinhard Zumkeller, Oct 05 2014

 a(n) = number of editing steps (replace, delete and insert) to transform n to A004086(n);
a(A002113(n)) = 0, a(10*A002113(n)) = 1 for n > 0;
a(A248336(n)) = n and a(m) != n for m < A248336(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Haskell Wiki, Edit distance
WikiBooks: Algorithm Implementation, Levenshtein distance
Wikipedia, Levenshtein distance

Cf. A004086, A002113, A248336.

a248327 0 = 0
a248327 n = levenshtein (show n) (dropWhile (== '0') $ reverse $ show n)
levenshtein :: (Eq t) => [t] -> [t] -> Int
levenshtein us vs = last $ foldl transform [0..length us] vs where
   transform xs@(x:xs') c = scanl compute (x+1) (zip3 us xs xs') where
      compute z (c', x, y) = minimum [y+1, z+1, x + fromEnum (c' /= c)]

A269588 Numbers n such that n^2 ends with the digits of n reversed (A004086(n)).

Original entry on oeis.org

1, 5, 6, 963, 9867, 65766, 69714, 6317056, 90899553, 169605719, 4270981082, 96528287587, 465454256742, 692153612536, 182921919071841, 655785969669834, 650700037578750084, 125631041500927357539, 673774165549097456624, 16719041449406813636569

Offset: 1

José Eduardo Gaboardi de Carvalho, Mar 01 2016

a(29)>10^32 (if it exists)

			6317056^2 = 39905196507136 which ends with 6507136, so 6317056 is a term.

Robert Gerbicz, Table of n, a(n) for n = 1..28
IBM, "Ponder This" puzzle for March 2016

Subsequence of A115761.

Cf. A003226, A004086.

Select[Range[10^7], Function[k, Take[IntegerDigits[#^2], -Length@ k] == Reverse@ k]@ IntegerDigits@ # &] (* Michael De Vlieger, Mar 04 2016 *)

isA269588(n)=dn = digits(n); rn = subst(Polrev(dn), x, 10); nbd = #dn; (n^2 - rn) % 10^nbd == 0; \\ Michel Marcus, Mar 01 2016

\\ printA269588len(d) prints all terms of the sequence with d digits
rev(n) = eval(concat(Vecrev(Str(n))));
{ printA269588len(d) = my(l, u, n); l=ceil(d/2); u=floor(d/2); for(y=0, 10^l-1, n=rev(y^2 % 10^u)*10^l+y; if(#Str(n)==d && Mod(n, 10^d)^2==rev(n), print(n)); ); }
\\ Max Alekseyev, Mar 07 2016

a(18)-a(20) from Max Alekseyev, Mar 07 2016
a(21)-a(27) from Robert Gerbicz, Apr 03 2016
a(28) from Dieter Beckerle, Jun 09 2016

cp's OEIS Frontend

A077741 Smallest multiple of n which begins with R(n) and ends in n where R(n) (A004086) is the digit reversal of n. Suitable number of zeros are assumed to the left of the MSD if required.

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A100414 Numbers n such that n is R(n)-th composite number where R(n) is the digit reversal of n (A002808(A004086(n))=n).

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A100415 Numbers n such that n is R(n)-th nonprime number, where R(n) is the digit reversal of n (A018252(A004086(n))=n).

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A104154 For each natural number n: if the last digit of n is not zero and A004086(n) is prime, append A004086(n) to the sequence.

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

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A165696 a(n) is the smallest number m such that the n numbers A004086(m^k) (digit reversal of m^k) for 0 < k < n+1 are all primes.

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Extensions

A209914 Number of ways n can be written as a multiple of its reversal A004086(n) +/- a prime p < n.

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PARI

A237912 Smallest number m (not ending in a 0) such that m and its digit reversal A004086(m) both have n prime factors (counted with multiplicity).

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Python

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A248327 Levenshtein distance of n and its reversal in decimal representation, cf. A004086.

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