A169824 -id:A169824 - cp's OEIS frontend

A031877 Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.

8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901, 87128712, 87999912, 98019801, 98999901, 871208712, 879999912, 980109801, 989999901, 8712008712, 8791287912, 8799999912, 9801009801, 9890198901, 9899999901, 87120008712, 87912087912, 87999999912

Offset: 1

Eric W. Weisstein

The terms of this sequence are sometimes called palintiples.
All terms are of the form 87...12 = 4*21...78 or 98...01 = 9*10...89. [This was proved by Hoey, 1992. - N. J. A. Sloane, Oct 19 2014] More precisely, they are obtained from concatenated copies of either 8712 or 9801, with 9's inserted "in the middle of" these and/or 0's inserted between the copies these, in a symmetrical way. A008919 lists the reversals, but not in the same order, e.g., R(a(2)) < R(a(1)). - M. F. Hasler, Aug 18 2014
There are 2*Fibonacci(floor((n-2)/2)) terms with n digits (this is A214927 or essentially twice A103609). - Ray Chandler, Oct 11 2017

W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 14-15, 1987.
G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104-105 (describes this problem as having "nothing in [it] which appeals much to a mathematician.").

Ray Chandler, Table of n, a(n) for n = 1..10000
Martin Beech, A Computer Conjecture of a Non-Serious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 50-51.
Patrick De Geest, Palindromic Products of Integers and their Reversals
D. J. Hoey, Palintiples
D. J. Hoey, Palintiples [Cached copy]
Benjamin V. Holt, Some General Results and Open Questions on Palintiple Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014.
Benjamin V. Holt, A Determination of Symmetric Palintiples, arXiv:1410.2356 [math.NT], 2014.
Benjamin V. Holt, Families of Asymmetric Palintiples Constructed from Symmetric and Shifted-Symmetric Palintiples, arXiv:1412.0231 [math.NT], 2014.
L. H. Kendrick, Young Graphs: 1089 et al, arXiv:1410.0106 [math.NT], 2014.
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.
Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132.
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
Eric Weisstein's World of Mathematics, Reversal.

See A008919 for reversals (this is the main entry for the problem).
Cf. A169824, A214927, A103609.
Union of A222814 and A222815.
Subsequence of A118959.

a031877_list = [x | x <- [1..], x `mod` 10 > 0,
                    let x' = a004086 x, x' /= x && x `mod` x' == 0]
-- Reinhard Zumkeller, Jul 15 2013

fQ[n_] := Block[{id = IntegerDigits@n}, Mod[n, FromDigits@ Reverse@id] == 0 && n != FromDigits@ Reverse@ id && Mod[n, 10] > 0]; k = 1; lst = {}; While[k < 10^9, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Jun 11 2010 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[Flatten[ {(4*198)#,(9*99)#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0,1},n], okQ],{n,12}]]]] (* Harvey P. Dale, Jul 03 2013 *)

is_A031877(n)={n%10 && n%A004086(n)==0 && n>A004086(n)} \\ M. F. Hasler, Aug 18 2014

A031877 = []
for n in range(1,10**7):
    if n % 10:
        s1 = str(n)
        s2 = s1[::-1]
        if s1 != s2 and not n % int(s2):
            A031877.append(n) # Chai Wah Wu, Sep 05 2014

a(n) = A004086(a(n))*[9/(a(n)%10)], where [...]=9 if a(n) ends in "1" and [...]=4 if a(n) ends in "2". - M. F. Hasler, Aug 18 2014

More terms from Jud McCranie, Aug 15 2001

More terms from Sam Mathers, Aug 18 2014

A281625 Numbers m>0 such that m = k(reversal of km) for some k<=m.

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

Offset: 1

Jaroslav Krizek, Feb 11 2017

Generalization of palindrome numbers in base 10.
Sequence is not the same as A061917 or A169824; a(188) = 3267 is not a term of these sequences.
Supersequence of A002113 (palindromes in base 10) and A061917.
Sequences a(n)_k of numbers m such that m = k*(reversal of k*m) for k <= 30 and n >= 1:
a(n)_1 = A002113(n+1) (palindromes > 0 in base 10);
a(n)_2 = 4356, 43956, 439956, 4399956, 43999956, 439999956, ...;
a(n)_3 = 3267, 32967, 329967, 3299967, 32999967, 329999967, ...;
a(n)_5 = a(n)_20 = 10*a(n)_2 = 43560, 439560, 4399560, 43999560, ...;
a(n)_8 = 6600, 6606600, 66006600, 660006600, ...;
a(n)_10 = 10*A002113(n+1): 10, 20, 30, 40, 50, 60, 70, 80, 90, 110, ... ;
a(n)_30 = 10*a(n)_3 = 32670, 329670, 3299670, 32999670, ...

			3267 is in the sequence because 3267 = 3*(reversal of 3*3267) = 3*(reversal of 9801) = 3*1089.

Jaroslav Krizek, Table of n <= 5000

Cf. A061917, A002113, A169824.

[n: k in [1..n], n in [1..1000] | n eq k * Seqint(Reverse(Intseq(k*n)))];

read("transforms") :
isA281625 := proc(n)
    for k from 1 to n do
        if k*digrev(k*n) = n then
            return true ;
        end if;
    end do:
    false;
end proc:
A281625 := proc(n)
    option remember ;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA281625(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A281625(n),n=1..100) ; # R. J. Mathar, Aug 06 2019

Select[Range@ 353, Function[n, Total@ Boole@ Map[Function[k, n == k FromDigits@ Reverse[IntegerDigits[k n]]], Range@ n] > 0]] (* Michael De Vlieger, Feb 11 2017 *)

A345361 Numbers that are not palindromes even after removing trailing zeros and are divisible by their reverses.

Original entry on oeis.org

510, 540, 810, 2100, 4200, 5100, 5200, 5400, 5610, 5700, 5940, 6300, 8100, 8400, 8712, 8910, 9801, 21000, 23100, 27000, 42000, 46200, 51000, 51510, 52000, 52200, 52800, 54000, 54540, 56100, 56610, 57000, 57200, 59400, 59940, 63000, 65340, 69300, 81000, 81810, 84000, 87120

Offset: 1

Tanya Khovanova, Jun 16 2021

Palindromes with or without trailing zeros are trivially divisible their reverses.

If k is in the sequence then 10*k is in the sequence. - David A. Corneth, Jun 16 2021

			510 is divisible by its reverse 15. Thus, 510 is in this sequence.

Cf. A002113, A004086, A169824.

def pal(s): return s == s[::-1]
def rmz(s): return s.strip('0')
def ok(n): s = str(n); return not (pal(s) or pal(rmz(s)) or n%int(s[::-1]))
print(list(filter(ok, range(82000)))) # Michael S. Branicky, Jun 16 2021

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A031877 Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.

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A281625 Numbers m>0 such that m = k(reversal of km) for some k<=m.

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A345361 Numbers that are not palindromes even after removing trailing zeros and are divisible by their reverses.

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cp's OEIS Frontend

A031877 Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

A281625 Numbers m>0 such that m = k*(reversal of k*m) for some k<=m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A345361 Numbers that are not palindromes even after removing trailing zeros and are divisible by their reverses.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

A281625 Numbers m>0 such that m = k(reversal of km) for some k<=m.