Re: Numbers and their reversals

*To*: mathgroup at smc.vnet.net*Subject*: [mg53715] Re: [mg53687] Numbers and their reversals*From*: DrBob <drbob at bigfoot.com>*Date*: Mon, 24 Jan 2005 03:37:47 -0500 (EST)*References*: <200501230702.CAA11076@smc.vnet.net> <opsk21ldpjiz9bcq@monster.ma.dl.cox.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

By the way, 91 isn't the most frequent factor in these problems. Here are the top ten factors for products up to six digits (with counts): Timing@Take[Sort@Frequencies@Flatten[omariD /@ Range@999999], -10] {46.797*Second, {{286, 182}, {605, 22}, {676, 4}, {891, 1001}, {1151, 111}, {1951, 91}, {2921, 101}, {3969, 3}, {15376, 2}, {16593, 11}}} 91 appears in 1951 cases. Bobby On Sun, 23 Jan 2005 19:11:27 -0600, DrBob <drbob at bigfoot.com> wrote: > (Some post as before, but there may have been a bad character in place of "!=".) > > There are actually LOTS of solutions. > > Here's a verification of the ones you listed, including OTHER factors that work (other than 17 and 91). I'm not counting trivial cases where the product is its own reverse. > > Clear[omariQ, omariD] > reverse = FromDigits@Reverse@IntegerDigits@# &; > commonDivisors = Rest@Divisors@GCD[#, reverse@#] &; > omariQ[n_Integer, > k_Integer] := n != reverse@n && IntegerQ[n/k] && reverse[n/k] == reverse[n]/k > omariD[n_Integer] := Select[commonDivisors@n, omariQ[n, #] &] > omariQ[n_Integer] := Length@omariD@n > 0 > omariD /@ {45628, 271726, 82654, 627172} > omariQ /@ {45628, 271726, 82654, 627172} > > {{17,34,187,374},{91},{17,34,187,374},{91}} > > {True,True,True,True} > > And this counts the 5-digit numbers that can be factored that way: > > Timing@Length@Select[Range[10000, 99999], omariQ] > > {4.5 Second, 5887} > > That's more than 6.5% of numbers in the range. > > If we want only those for which 17 can be the common factor, here they are up to 5 digits: > > Timing@Select[Range@99999, omariQ[#, 17] &] > > {1.296 Second, {41140, 45628, 82280, 82654}} > > 91 yields quite a few more choices: > > Timing@Select[Range@99999, omariQ[#, 91] &] > > {1.25 Second, {10010, 11102, > 12012, 12103, 13013, 13104, 14014, 14105, 15015, 15106, 16016, 16107, > 17017, 17108, 18018, 18109, 20020, 20111, 21021, 21203, 22113, 22204, > 23023, 23114, 23205, 24024, 24115, 24206, 25025, 25116, 25207, 26026, > 26117, 26208, 27027, 27118, 27209, 30030, 30121, 30212, 31031, 31122, > 31304, 32032, 32214, 32305, 33124, 33215, 33306, 34034, 34125, 34216, > 34307, 35035, 35126, 35217, 35308, 36036, 36127, 36218, 36309, 40040, > 40131, 40222, 40313, 41041, 41132, 41223, 41405, 42042, 42133, 42315, > 42406, 43043, 43225, 43316, 43407, 44135, 44226, 44317, 44408, 45045, > 45136, 45227, 45318, 45409, 50050, 50141, 50232, 50323, 50414, 51051, > 51142, 51233, 51324, 51506, 52052, 52143, 52234, 52416, 52507, 53053, > 53144, 53326, 53417, 53508, 54054, 54236, 54327, 54418, 54509, 60060, > 60151, 60242, 60333, 60424, 60515, 61061, 61152, 61243, 61334, 61425, > 61607, 62062, 62153, 62244, 62335, 62517, 62608, 63063, 63154, 63245, > 63427, 63518, 63609, 70070, 70161, 70252, 70343, 70434, 70525, 70616, > 71071, 71162, 71253, 71344, 71435, 71526, 71708, 72072, 72163, 72254, > 72345, 72436, 72618, 72709, 80080, 80171, 80262, 80353, 80444, 80535, > 80626, 80717, 81081, 81172, 81263, 81354, 81445, 81536, 81627, 81809, > 90090, 90181, 90272, 90363, 90454, 90545, 90636, 90727, 90818}} > > Bobby > > On Sun, 23 Jan 2005 02:02:17 -0500 (EST), F. omari <towtoo2002 at yahoo.com> wrote: > >> >> i want to investigate the following two equations: >> a * const = z >> a_Reversed * const = z_Reversed >> where a, z, and their reversed form and const are all positive integers >> ie such that: >> 2684 * 17 = 45628 >> 4862 * 17 = 82654 >> 2986 * 91 = 271726 >> 6892 * 91 = 627172 >> it happened that many multipliers of 91 have such a property. >> while the multipliers of 17 have only 5 cases in the interval of 1 to 3000 >> the following code will investigate the multipliers of 17, to investigate another number just replace 17. and you may increase the interval of investigation. i am sure that my code is an old fashion one, please any other ideas about a more functional code. >> a = Table[i, {i, 1, 3000}]; zR = ""; aR = 0; z = ""; >> Do[aR = ToExpression[StringReverse[ToString[a[[i]]]]]; >> z = ToString[a[[i]]*17]; >> zR = StringReverse[ToString[aR*17]]; >> If[zR == z, Print[a[[i]]]], {i, 1, 3000}] >> >> 242 >> 484 >> 2442 >> 2662 >> 2684 >> regards >> >> >> >> >> > > > -- DrBob at bigfoot.com www.eclecticdreams.net

**References**:**Numbers and their reversals***From:*"F. omari" <towtoo2002@yahoo.com>