A031877 -id:A031877 - cp's OEIS frontend

A091082 Numbers n which when converted to base 8, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

42, 378, 2625, 2730, 3066, 3402, 3969, 21546, 23625, 24570, 27594, 32193, 170625, 172074, 174762, 191625, 193914, 196602, 221130, 257985, 1346625, 1376298, 1400490, 1535625, 1548666, 1572858, 1769418, 2064321, 10754625, 10922625, 11010090, 11031594, 11184810

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

			a(1) = 42 because: 42 in base 8 is 52; 52 reversed is 25; 25 converted back to base 10 is 21 and 42 mod 21 = 0.

C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link]

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091080 (base 6), A091081 (base 7), A091083 (base 9), A031877 (base 10).

/* See A091077 and use PARI script with b=8 */

More terms from Michel Marcus, Oct 10 2014

A091083 Numbers n which when converted to base 9, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

64, 640, 4800, 5248, 5824, 6400, 43680, 46720, 52480, 58240, 393600, 419968, 425152, 467200, 472384, 524800, 3542880, 3779200, 3831040, 4199680, 4251520, 4723840, 31497600, 31886400, 34012288, 34058944, 34437376, 34484032, 37792000, 37838656, 38217088

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

			a(1) = 64 because 64 in base 9 is 71; 71 reversed is 17; 17 converted back to base 10 is 16 and 64 mod 16 = 0.

C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link]

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091080 (base 6), A091081 (base 7), A091082 (base 8), A031877 (base 10).

/* See A091077 and use PARI script with b=9 */

More terms from Michel Marcus, Oct 10 2014

A118959 Non-palindromic numbers which are divisible by their reversal.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 200, 220, 300, 330, 400, 440, 500, 510, 540, 550, 600, 660, 700, 770, 800, 810, 880, 900, 990, 1000, 1010, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2000, 2020, 2100, 2120, 2200, 2220, 2320, 2420

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 25 2006

A004086(a(n)) = A027751(a(n),k) for some k: 1 <= k < A001221(n); A031877 contains all terms not ending with zero. - Reinhard Zumkeller, Jul 15 2013

			510 is in the sequence because 510 is a non-palindromic number divisible by its reversal 15.

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

Subsequence of A029742; A031877 is a subsequence.

a118959 n = a118959_list !! (n-1)
a118959_list = filter
   (\x -> let x' = a004086 x in x' /= x && x `mod` x' == 0) [1..]
-- Reinhard Zumkeller, Jul 15 2013

Select[Range[2420],!PalindromeQ[#]&&Divisible[#,IntegerReverse[#]]&] (* James C. McMahon, Sep 13 2024 *)

A083406 Even squares which can be expressed as the product of a number and its reversal in at least two different ways.

Original entry on oeis.org

63504, 435600, 6350400, 7683984, 16240900, 25401600, 43560000, 66585600, 420332004, 558471424, 635040000, 647804304, 726949444, 768398400, 782432784, 1067328900, 1624090000, 1897473600, 2341011456, 2540160000

Offset: 1

Shyam Sunder Gupta, Jun 07 2003

For n=1..49 identical to A083408.

			63504 = 252 * 252 = 144 * 441, 7683984 = 2772 * 2772 = 1584 * 4851, etc.

S. S. Gupta, EPRNs, Science Today, Feb. 1987, India.

Chai Wah Wu, Table of n, a(n) for n = 1..2372 (first 73 terms from Hans Havermann)
Shyam Sunder Gupta, Equal Product of Reversible Numbers

Cf. A031877, A066531, A083407 (odd squares version), A083408 (all squares version).

Corrected and extended by Hans Havermann, Feb 11 2012

Definition corrected by N. J. A. Sloane, Aug 01 2019

A169824 Numbers n such that n is divisible by n-with-its-digits-reversed.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 300, 303, 313, 323, 330, 333, 343, 353, 363, 373, 383, 393

Offset: 1

N. J. A. Sloane, May 29 2010

If n is a term, then so is n*10^k. Positive terms in A002113 is a subsequence. - Chai Wah Wu, Sep 28 2017

a(n) = A061917(n+1) for 1 <= n < 78, but a(78) = 510 differs from A061917(79) = 515. - Georg Fischer, Oct 28 2018

			40 is divisible by 4.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Erich Friedman, What's Special About This Number? (See the entry for "540")
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.

Cf. A031877.

Cf. A002113. - Robert G. Wilson v, Jun 10 2010

fQ[n_] := Mod[n, FromDigits@ Reverse@ IntegerDigits@ n] == 0; Select[ Range@ 399, fQ@# &] (* Robert G. Wilson v, Jun 10 2010 *)

A169824_list = [n for n in range(1,1000) if not n % int(str(n)[::-1])] # Chai Wah Wu, Sep 28 2017

A071685 Non-palindromic numbers n, not divisible by 10, such that either n divides R(n) or R(n) divides n, where R(n) is the digit-reversal of n.

Original entry on oeis.org

1089, 2178, 8712, 9801, 10989, 21978, 87912, 98901, 109989, 219978, 879912, 989901, 1099989, 2199978, 8799912, 9899901, 10891089, 10999989, 21782178, 21999978, 87128712, 87999912, 98019801, 98999901, 108901089, 109999989

Offset: 1

Labos Elemer, Jun 03 2002

The quotient R(n)/n or n/R(n) is always 4 or 9.
This is the union of the four sequence A001232, A222814, A008918, A222815. Equivalently, the union of A008919 and A031877.
There are 4*Fibonacci(floor((n-2)/2)) terms with n digits (this is 2*A214927 or essentially 4*A103609). - Ray Chandler, Oct 12 2017
Conjecture: every term mod 100 is equal to 1, 12, 78, or 89. - Harvey P. Dale, Dec 13 2017

			Palindromic solutions like 12021 or also solutions divisible by 10 were filtered out like {8380,838; q=10} or {8400,48; q=175}. In case of m>R(m), q=m/R(m)=4 or 9.

Ray Chandler, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, 2178 And All That, arXiv:1307.0453 [math.NT], 2013; Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Cf. A001232, A222814, A008918, A222815, A008919, A031877.

Cf. A004086, A055483, A069554, A103609, A214927.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[IntegerDigits[x]] Do[s=Mod[Max[{n, tn[red[n]]}], Min[{n, r=tn[red[n]]}]]; If[Equal[s, 0]&&!Equal[Mod[n, 10], 0] &&!Equal[n, r], Print[{n, r/n}]], {n, 1, 1000000}]
npnQ[n_]:=Module[{r=IntegerReverse[n]},!PalindromeQ[n]&&!Divisible[ n,10] &&(Mod[n,r]==0||Mod[r,n]==0)]; Select[Range[11*10^7],npnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2017 *)

x = q*R(x), q is an integer q<>1, q<>10^j and neither of x or R(x) is divisible by 10.

Corrected and extended by Harvey P. Dale, Jul 01 2013
Edited by N. J. A. Sloane, Jul 02 2013
Missing terms inserted by Ray Chandler, Oct 09 2017
Incorrect comment removed by Ray Chandler, Oct 12 2017

A082945 Numbers n which in decimal have the form imj, where m is the middle digit, with property that j is the reversal of i, and i = m*j.

Original entry on oeis.org

111, 212, 313, 414, 515, 616, 717, 818, 919, 11111, 22122, 33133, 44144, 55155, 66166, 77177, 88188, 99199, 1011101, 1111111, 1211121, 1311131, 1411141, 1511151, 1611161, 1711171, 1811181, 1911191, 2021202, 2121212, 2221222, 2321232, 2421242

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 07 2003

The number i should not end in 0, and m should not equal 0.
Suggested by Amarnath Murthy.
Either i and j are identical palindromes and m is 1, or i is in A031877, j is the corresponding term in A008919, and m is either 4 or 9. - Charlie Neder, Mar 08 2019

			rev(8712) = 2178 and 8712 = 2178*4, so 871242178 is in this sequence. - _Charlie Neder_, Mar 08 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002113, A008919, A031877. (Possible values of i and j)

Edited by N. J. A. Sloane, Oct 25 2009, following discussions on the Sequence Fans Mailing List, circa Apr 17 2009

Corrected and extended by Charlie Neder, Mar 08 2019

A243097 Digit sums of the nontrivial reversal numbers (numbers which are integer multiples of their reversal), excluding palindromes and multiples of 10.

Original entry on oeis.org

18, 18, 27, 27, 36, 36, 45, 45, 36, 54, 36, 54, 36, 63, 36, 63, 36, 54, 72, 36, 54, 72, 36, 54, 81, 36, 54, 81, 36, 54, 54, 72, 90, 36, 54, 54, 72, 90, 36, 63, 54, 72, 99, 36, 63, 54, 72, 99, 36, 54, 72, 54, 72, 72, 90, 108, 36, 54, 72, 54, 72, 72, 90, 108

Offset: 1

Sam Mathers, Aug 18 2014

			a(1)=digit sum of 8712=8+7+1+2=18, a(3)=digit sum of 87912=8+7+9+1+2=27.

Cf. A007953 (digit sum), A031877 (union of A222814 and A222815).

A243097 = []
for n in range(1,10**7):
    if n % 10:
        s1 = str(n)
        s2 = s1[::-1]
        if s1 != s2 and not n % int(s2):
            A243097.append(sum(int(d) for d in s1))
# Chai Wah Wu, Sep 05 2014

a(n) = 9(d-2p-b) where d is the number of digits in the reversal number, p is the number of repeating units of either 8712 or 9801 (they can be split in the middle of the sequence as long as all 4 numbers appear in the correct order), and b is the number of digits separating two complete "units" (must appear between two complete units and not in between 1 incomplete unit).

a(n) = A007953(A031877(n)).

More terms from Michel Marcus, Aug 25 2014

A338545 Numbers that escape to infinity when applying this algorithm: if x_i <= rev(x_i) then x_i+1 = rev(x_i) - x_i else x_i+1 = rev(x_i) + x_i.

Original entry on oeis.org

3763, 3853, 3943, 3973, 4492, 4582, 4672, 4762, 4792, 4852, 4882, 4942, 4972, 5194, 5284, 5374, 5464, 5491, 5554, 5581, 5644, 5671, 5734, 5761, 5791, 5824, 5851, 5881, 5914, 5941, 5971, 5993, 6193, 6283, 6373, 6426, 6463, 6490, 6516, 6553

Offset: 1

Ruediger Jehn, Nov 02 2020

All numbers up to 3762 (not already being a palindrome) are mapped sooner or later onto a palindrome and collapse to 0 in the next step of the algorithm. But unlike in the Collatz conjecture (see A006370) where probably all numbers fall back to 1, here most numbers never fall back to 0. Instead, they fall onto repetitive patterns which increase by one order of magnitude every three steps and hence go to infinity. For instance if the algorithm reaches the number 19799999999999991002000000000000088 at step k it will reach 1979999999999999910020000000000000088 at step k+6. And every six steps one 9 is added to the long line of 9's and one 0 is added right of the 2. Up to 100000 there are 3 repetitive patterns, which contain the numbers listed in the second line of the Python code below (escape = [...]).
From the first 1 million numbers, 489806 fall back to zero whereas the other 510194 fall into one of 6 repetitively increasing patterns.
The terms become more and more frequent: up to 10000 they just make 1.78 %, up to 100000 they are 21.88 % and up to 1000000 they are already 51.02 %.
Many numbers (like the 137 in the example below) hit at some point the numbers 8712 and 9801 that are terms of A031877, which is not a coincidence. They fall back to 0.
338545 is a term of this sequence A338545, hence it is also a term of A053873.

			x_0 = 137, x_1 = 731 - 137 = 594, x_2 = 495 + 594 = 1089, x_3 = 9801 - 1089 = 8712, x_4 = 2178 + 8712 = 10890, x_5 = 9801 + 10890 = 20691, x_6 = 19602 + 20691 = 40293, x_7 = 39204 + 40293 = 79497, x_8 = 79497 - 79497 = 0, hence 137 is not a term.

Cf. A031877.

sequence = []
escape = [1090089, 99100089, 9900109899999990109989]
for k in range(1, 100000):
    x = k
    while not x==0:
       if x <= rev(x):
          x = rev(x) - x
       else:
          x = rev(x) + x
       if x in escape:
          x = 0
          sequence.append(k)

A100475 Prime-th recurrence with reversal at each step.

Original entry on oeis.org

1, 2, 3, 5, 11, 13, 14, 34, 931, 7037, 95017, 3549221, 75296795, 3518502151, 34778126848, 747746329129, 13428156340222, 728260738568834, 17205012007789762

Offset: 0

Jonathan Vos Post, Nov 22 2004

Starting at other than a(n) = 1, does this sequence ever go into a loop?

			a(7) = 14 because a(6) = 13, the 13th prime is 41 and reversal(41) = 14.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 14-15, 1987.

Andrew Booker, The Nth Prime Page
Eric Weisstein's World of Mathematics, Reversal.
Eric Weisstein's World of Mathematics, Rosser's Theorem.

Cf. A007097, A031877.

f[n_] := FromDigits[ Reverse[ IntegerDigits[ Prime[n]]]]; NestList[f, 1, 16] (* Robert G. Wilson v, Dec 02 2004 *)
NestList[IntegerReverse[Prime[#]]&,1,19] (* Harvey P. Dale, May 02 2022 *)

a(n) = reversal(a(n-1)th prime).

a(15)-a(16) from Robert G. Wilson v, Dec 02 2004

a(17)-a(18) from Paul Zimmermann, Dec 04 2004, computed thanks to Deléglise-Rivat's program; confirmed by Marc Deléglise, Dec 09 2004

cp's OEIS Frontend

A091082 Numbers n which when converted to base 8, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

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A091083 Numbers n which when converted to base 9, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

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A118959 Non-palindromic numbers which are divisible by their reversal.

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A083406 Even squares which can be expressed as the product of a number and its reversal in at least two different ways.

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A169824 Numbers n such that n is divisible by n-with-its-digits-reversed.

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A071685 Non-palindromic numbers n, not divisible by 10, such that either n divides R(n) or R(n) divides n, where R(n) is the digit-reversal of n.

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A082945 Numbers n which in decimal have the form imj, where m is the middle digit, with property that j is the reversal of i, and i = m*j.

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A243097 Digit sums of the nontrivial reversal numbers (numbers which are integer multiples of their reversal), excluding palindromes and multiples of 10.

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