A118959 -id:A118959 - 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

A061467 Remainder when the larger of n and its reverse is divided by the smaller.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 5, 13, 6, 13, 3, 9, 15, 0, 9, 0, 9, 18, 2, 10, 18, 26, 5, 0, 5, 9, 0, 9, 18, 27, 36, 7, 15, 0, 13, 18, 9, 0, 9, 18, 27, 36, 45, 0, 6, 2, 18, 9, 0, 9, 18, 27, 36, 0, 13, 10, 27, 18, 9, 0, 9, 18, 27, 0, 3, 18, 36, 27, 18, 9, 0, 9, 18, 0, 9, 26, 7, 36, 27

Offset: 0

Erich Friedman, Jun 16 2001

a(n)=0 if n is in A002113, A008919 or A118959. - Robert Israel, Jul 18 2019

			a(12)=9 since 21/12 = 1 with remainder 9.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A002113, A004086, A008919, A118959.

a061467 0 = 0
a061467 n = mod (max n n') (min n n') where n' = a004086 n
-- Reinhard Zumkeller, Dec 31 2013

l := {} For[i = 1, i < 100, i++, x := FromDigits[Reverse[IntegerDigits[i]]]; If[x >= i, AppendTo[l, Mod[x, i]], AppendTo[l, Mod[i, x]]]] l (* Jake Foster, Jun 05 2008 *)
rln[n_]:=Module[{r=IntegerReverse[n]},If[r>n,Mod[r,n],Mod[n,r]]]; Join[ {0}, Array[rln,90]] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Apr 03 2016 *)

{ for (n=0, 1000, x=n; r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); p=max(n, r); q=min(n, r); write("b061467.txt", n, " ", p%q) ) } \\ Harry J. Smith, Jul 23 2009

A087993 Numbers k for which the quotient q(k)=(k+rev(k))/abs(k-rev(k)) is an integer.

Original entry on oeis.org

45, 54, 495, 594, 4356, 4545, 4995, 5454, 5895, 5985, 5994, 6534, 10890, 19602, 20691, 29403, 30492, 39204, 40293, 43956, 45045, 49005, 49995, 50094, 54054, 59994, 65934, 68607, 70686, 77319, 91377, 109890, 197802, 208791, 296703, 307692

Offset: 1

Labos Elemer, Oct 08 2003

			n = 45, rev(n) = 54, q = 99/9 = 9 integer;
n = 934065, rev(n) = 560434, q = 1494504/373626 = 4, integer.

Giovanni Resta, Table of n, a(n) for n = 1..500

Cf. A087994 (the quotients), A118959.

rev[n_] := FromDigits@ Reverse@ IntegerDigits@ n; Select[Range[10^5], # != (r = rev[#]) && Mod[# + r, # - r] == 0 &] (* Giovanni Resta, May 14 2017 *)

A273239 Non-palindromic numbers whose reversal is a palindrome.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 200, 220, 300, 330, 400, 440, 500, 550, 600, 660, 700, 770, 800, 880, 900, 990, 1000, 1010, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2000, 2020, 2120, 2200, 2220, 2320, 2420, 2520, 2620, 2720, 2820, 2920

Offset: 1

Giovanni Teofilatto, May 18 2016

Subsequence of A118959. - Altug Alkan, May 18 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A118959, A273245.

More terms from Altug Alkan, May 18 2016

A359173 Numbers whose square can be expressed as k * A004086(k) with non-palindromic k.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 200, 220, 252, 300, 330, 400, 403, 440, 500, 504, 550, 600, 660, 700, 770, 800, 816, 880, 900, 990, 1000, 1010, 1100, 1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910, 2000, 2020, 2120, 2200, 2220, 2320, 2420, 2520, 2620, 2720, 2772

Offset: 1

Hugo Pfoertner, Dec 17 2022

If k is a term, then so is 10*k. - Robert Israel, Dec 23 2022

			a(1) = 10 because 100*1 = 10^2;
a(2) = 20: 200*2 = 20^2;
a(11) = 110: 1100*11 = 110^2;
a(14) = 252: 144*441 = 252^2;
a(28) = 816: 768*867 = 816^2.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000290, A002113, A004086, A118959.

rev:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc:
g:= proc(d,m) local r; r:= rev(d); r <> d and m = d*r end proc:
filter:= proc(n) ormap(g, numtheory:-divisors(n^2),n^2) end proc:
select(filter, [$1..3000]); # Robert Israel, Dec 23 2022

L=List(); for (k=1, 3*10^6, my (r=fromdigits(Vecrev(digits(k))), s); if (issquare(r*k, &s) && r!=k, if(s<3001, listput(L, s)))); Set(L)

from itertools import count, islice
from sympy import divisors
def A359173_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:any(d*int(str(d)[::-1])==n**2 for d in divisors(n**2,generator=True) if d != n),count(max(startvalue,1)))
A359173_list = list(islice(A359173_gen(),30)) # Chai Wah Wu, Dec 19 2022

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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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A061467 Remainder when the larger of n and its reverse is divided by the smaller.

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A087993 Numbers k for which the quotient q(k)=(k+rev(k))/abs(k-rev(k)) is an integer.

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Mathematica

A273239 Non-palindromic numbers whose reversal is a palindrome.

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A359173 Numbers whose square can be expressed as k * A004086(k) with non-palindromic k.

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