[math-fun] a(n) and a(n+1) substrings of their product
Hello Math-Fun, Carole Dubois and myself have tried to extend S -- but could not find a new term < 9 999 999 999. S = 1, 10, 11, 100, 21, 1000, 31, 10000, 41, 100000, 51, 1000000, 61, 10000000, 71, 100000000, 81, 2817. S should be the lexicographically earliest seq of distinct positive terms such that a(n) and a(n+1) are substrings of their product. Example: 81 * 2817 = 228177. Does S end there? Would a different a(1) produce more terms? Is there another way to exploit this idea of two terms visible in their product? (respectively addition, division) Or is this old hat? Best, É.
old hat? ... mmmmh, there is something there: http://oeis.org/A066217
Le 23 févr. 2020 à 00:30, Éric Angelini <eric.angelini@skynet.be> a écrit :
Hello Math-Fun, Carole Dubois and myself have tried to extend S -- but could not find a new term < 9 999 999 999.
S = 1, 10, 11, 100, 21, 1000, 31, 10000, 41, 100000, 51, 1000000, 61, 10000000, 71, 100000000, 81, 2817.
S should be the lexicographically earliest seq of distinct positive terms such that a(n) and a(n+1) are substrings of their product. Example: 81 * 2817 = 228177. Does S end there? Would a different a(1) produce more terms? Is there another way to exploit this idea of two terms visible in their product? (respectively addition, division) Or is this old hat? Best, É.
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Hello Math-Fun, do you know any other solutions? (especially where A, B and C are distinct) 1818 = A 8182 = B + 8182 = C ------ = 18182 = D Best, É.
I did not make an exhaustive search, but the one you pointed to seems to be part of a series: 19 9 1 9 182 82 18 82 1819 819 181 819 18182 8182 1818 8182 181819 81819 18181 81819 1818182 818182 181818 818182 18181819 8181819 1818181 8181819 Frank On Mon, Feb 24, 2020 at 12:58 PM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, do you know any other solutions? (especially where A, B and C are distinct)
1818 = A 8182 = B + 8182 = C ------ = 18182 = D
Best, É.
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I ran a search out to a billion & found 109 examples. There were no cases of A, B and C distinct, and they pretty much all fell into a few simple patterns. Here's the list: 15=5+5+5 19=1+9+9 150=50+50+50 182=18+82+82 191=9+91+91 195=5+95+95 199=1+99+99 1500=500+500+500 1819=181+819+819 1950=50+950+950 1981=19+981+981 1991=9+991+991 1995=5+995+995 1999=1+999+999 15000=5000+5000+5000 18182=1818+8182+8182 19091=909+9091+9091 19500=500+9500+9500 19802=198+9802+9802 19901=99+9901+9901 19950=50+9950+9950 19981=19+9981+9981 19991=9+9991+9991 19995=5+9995+9995 19999=1+9999+9999 150000=50000+50000+50000 181819=18181+81819+81819 195000=5000+95000+95000 198020=1980+98020+98020 199010=990+99010+99010 199091=909+99091+99091 199500=500+99500+99500 199801=199+99801+99801 199901=99+99901+99901 199950=50+99950+99950 199981=19+99981+99981 199991=9+99991+99991 199995=5+99995+99995 199999=1+99999+99999 1500000=500000+500000+500000 1818182=181818+818182+818182 1909091=90909+909091+909091 1950000=50000+950000+950000 1980199=19801+980199+980199 1995000=5000+995000+995000 1998002=1998+998002+998002 1999001=999+999001+999001 1999010=990+999010+999010 1999091=909+999091+999091 1999500=500+999500+999500 1999801=199+999801+999801 1999901=99+999901+999901 1999950=50+999950+999950 1999981=19+999981+999981 1999991=9+999991+999991 1999995=5+999995+999995 1999999=1+999999+999999 15000000=5000000+5000000+5000000 18181819=1818181+8181819+8181819 19500000=500000+9500000+9500000 19801981=198019+9801981+9801981 19900991=99009+9900991+9900991 19909091=90909+9909091+9909091 19950000=50000+9950000+9950000 19980020=19980+9980020+9980020 19990010=9990+9990010+9990010 19995000=5000+9995000+9995000 19998001=1999+9998001+9998001 19999001=999+9999001+9999001 19999010=990+9999010+9999010 19999091=909+9999091+9999091 19999500=500+9999500+9999500 19999801=199+9999801+9999801 19999901=99+9999901+9999901 19999950=50+9999950+9999950 19999981=19+9999981+9999981 19999991=9+9999991+9999991 19999995=5+9999995+9999995 19999999=1+9999999+9999999 150000000=50000000+50000000+50000000 181818182=18181818+81818182+81818182 190909091=9090909+90909091+90909091 195000000=5000000+95000000+95000000 198019802=1980198+98019802+98019802 199009901=990099+99009901+99009901 199500000=500000+99500000+99500000 199800200=199800+99800200+99800200 199900100=99900+99900100+99900100 199900991=99009+99900991+99900991 199909091=90909+99909091+99909091 199950000=50000+99950000+99950000 199980002=19998+99980002+99980002 199990001=9999+99990001+99990001 199990010=9990+99990010+99990010 199995000=5000+99995000+99995000 199998001=1999+99998001+99998001 199999001=999+99999001+99999001 199999010=990+99999010+99999010 199999091=909+99999091+99999091 199999500=500+99999500+99999500 199999801=199+99999801+99999801 199999901=99+99999901+99999901 199999950=50+99999950+99999950 199999981=19+99999981+99999981 199999991=9+99999991+99999991 199999995=5+99999995+99999995 199999999=1+99999999+99999999 On Mon, Feb 24, 2020 at 3:58 AM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, do you know any other solutions? (especially where A, B and C are distinct)
1818 = A 8182 = B + 8182 = C ------ = 18182 = D
Best, É.
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I should say, I excluded 0 as an addend, because then you get a=a+0+0 for every a with a zero in it, which strikes me as not interesting. On Tue, Feb 25, 2020 at 10:26 AM Tom Duff <td@pixar.com> wrote:
I ran a search out to a billion & found 109 examples. There were no cases of A, B and C distinct, and they pretty much all fell into a few simple patterns. Here's the list: 15=5+5+5 19=1+9+9 150=50+50+50 182=18+82+82 191=9+91+91 195=5+95+95 199=1+99+99 1500=500+500+500 1819=181+819+819 1950=50+950+950 1981=19+981+981 1991=9+991+991 1995=5+995+995 1999=1+999+999 15000=5000+5000+5000 18182=1818+8182+8182 19091=909+9091+9091 19500=500+9500+9500 19802=198+9802+9802 19901=99+9901+9901 19950=50+9950+9950 19981=19+9981+9981 19991=9+9991+9991 19995=5+9995+9995 19999=1+9999+9999 150000=50000+50000+50000 181819=18181+81819+81819 195000=5000+95000+95000 198020=1980+98020+98020 199010=990+99010+99010 199091=909+99091+99091 199500=500+99500+99500 199801=199+99801+99801 199901=99+99901+99901 199950=50+99950+99950 199981=19+99981+99981 199991=9+99991+99991 199995=5+99995+99995 199999=1+99999+99999 1500000=500000+500000+500000 1818182=181818+818182+818182 1909091=90909+909091+909091 1950000=50000+950000+950000 1980199=19801+980199+980199 1995000=5000+995000+995000 1998002=1998+998002+998002 1999001=999+999001+999001 1999010=990+999010+999010 1999091=909+999091+999091 1999500=500+999500+999500 1999801=199+999801+999801 1999901=99+999901+999901 1999950=50+999950+999950 1999981=19+999981+999981 1999991=9+999991+999991 1999995=5+999995+999995 1999999=1+999999+999999 15000000=5000000+5000000+5000000 18181819=1818181+8181819+8181819 19500000=500000+9500000+9500000 19801981=198019+9801981+9801981 19900991=99009+9900991+9900991 19909091=90909+9909091+9909091 19950000=50000+9950000+9950000 19980020=19980+9980020+9980020 19990010=9990+9990010+9990010 19995000=5000+9995000+9995000 19998001=1999+9998001+9998001 19999001=999+9999001+9999001 19999010=990+9999010+9999010 19999091=909+9999091+9999091 19999500=500+9999500+9999500 19999801=199+9999801+9999801 19999901=99+9999901+9999901 19999950=50+9999950+9999950 19999981=19+9999981+9999981 19999991=9+9999991+9999991 19999995=5+9999995+9999995 19999999=1+9999999+9999999 150000000=50000000+50000000+50000000 181818182=18181818+81818182+81818182 190909091=9090909+90909091+90909091 195000000=5000000+95000000+95000000 198019802=1980198+98019802+98019802 199009901=990099+99009901+99009901 199500000=500000+99500000+99500000 199800200=199800+99800200+99800200 199900100=99900+99900100+99900100 199900991=99009+99900991+99900991 199909091=90909+99909091+99909091 199950000=50000+99950000+99950000 199980002=19998+99980002+99980002 199990001=9999+99990001+99990001 199990010=9990+99990010+99990010 199995000=5000+99995000+99995000 199998001=1999+99998001+99998001 199999001=999+99999001+99999001 199999010=990+99999010+99999010 199999091=909+99999091+99999091 199999500=500+99999500+99999500 199999801=199+99999801+99999801 199999901=99+99999901+99999901 199999950=50+99999950+99999950 199999981=19+99999981+99999981 199999991=9+99999991+99999991 199999995=5+99999995+99999995 199999999=1+99999999+99999999
On Mon, Feb 24, 2020 at 3:58 AM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, do you know any other solutions? (especially where A, B and C are distinct)
1818 = A 8182 = B + 8182 = C ------ = 18182 = D
Best, É.
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On 2020-02-25 10:28, Tom Duff wrote:
I should say, I excluded 0 as an addend, because then you get a=a+0+0 for every a with a zero in it, which strikes me as not interesting.
Adding 0's to a previous solution is also uninteresting. I assume you meant to cut those out, too, but you (mistakenly?) included all forms of 150* = 50* + 50* + 50* (but almost no other examples [e.g. 182, 1820, 18200, ...], although you do include 195000=5000+95000+95000 and 19500, and 195 = 5 + 95 + 95) To answer Eric's question, it is impossible to have 3 distinct values for A, B, C. Also, the leading digit of D must be 1, and the 2nd digit must be either 8 or 9 or (in special cases when D=A+A+A), 5. (Boring, grinding argument follows:) In this case, for D to be an n digit number, then it has to be the sum of at 2 n-1 digit numbers, or at least 3 n-1 digit numbers to have a leading digit higher than 1. There are only 2 distinct n-1 digit numbers possible (the first n-1 and the last n-1 digits of D) so let's restrict ourselves to D's with a leading 1 [actually, D *always* must have a leading 1]. Let's focus on, x, the 2nd digit of D. We'd need 1x... + x... + the n-2 digit number, which can start with at most 9. The 2nd digit, x, of D must be x = (x + 1 + carry from the lower digits) mod 10, but 1 + <carry> cannot = 10 (<carry> can't be > 2). So no such 2nd digit x exists. So no 3 distinct A, B, C exist. For the non-distinct case, either D = A + A + A, or D = 2*A + B In the latter case, A must be the last n-1 digits of D. (If the leading digit is either 1 or 2, then A can't be the first n-1 digits -- the 2nd digit x of D would have to be x = 4 + x + <carry> or x = 2 + x + <carry>. <carry> + either 2 or 4 can never = 10, so A must be the last n-1 digits of D, not the first). By the same sort of (by now boring) reasoning, it is easy to see that the leading digit of D must always be 1. (For leading digit 2, we'd need both x = 2*x + 2 + <carry> mod 10, and 2*x + 2 + <carry> >= 20, so 2*x + 2 + <carry> = 20 + x, or x + 2 + <carry> = 20, which is impossible (x is a single digit, and <carry> <= 2)) So leading digit of D is 1, and all solutions are of the form D = 2*A + B, where x is the leading digit of A and the 2nd digit of D. 2*x + 1 + <carry> = 10 + x, or 2*x + <carry> = 10 + x. (The former is if B is the n-1 leading digits of D, the latter is if B is any shorter set of digits of D). So x must be either 8 or 9 when D = 2*A + B The only time A = B = C is if the final digit of D is 0 or 5 (A cannot be the first n-1 digits of D, else 3 * the leading digit would have to equal itself). If any example exists with A and D ending in 0, D = A + A + A, then D/10 A/10 must also be an example. So all such examples have to have a single digit solution. 5 is the only such example. No 2 digit value for A ending in 5 is possible, because the 2nd digit x, must be = to the last digit of 3*x + 1. So 1500... = 500.. + 500.. + 500.. are the only such examples of A + A + A
On Tue, Feb 25, 2020 at 10:26 AM Tom Duff <td@pixar.com> wrote:
I ran a search out to a billion & found 109 examples. There were no cases of A, B and C distinct, and they pretty much all fell into a few simple patterns. Here's the list: 15=5+5+5 19=1+9+9 150=50+50+50 182=18+82+82 191=9+91+91 195=5+95+95 199=1+99+99 1500=500+500+500 1819=181+819+819 1950=50+950+950 1981=19+981+981 1991=9+991+991 1995=5+995+995 1999=1+999+999 15000=5000+5000+5000 18182=1818+8182+8182 19091=909+9091+9091 19500=500+9500+9500 19802=198+9802+9802 19901=99+9901+9901 19950=50+9950+9950 19981=19+9981+9981 19991=9+9991+9991 19995=5+9995+9995 19999=1+9999+9999 150000=50000+50000+50000 181819=18181+81819+81819 195000=5000+95000+95000 198020=1980+98020+98020 199010=990+99010+99010 199091=909+99091+99091 199500=500+99500+99500 199801=199+99801+99801 199901=99+99901+99901 199950=50+99950+99950 199981=19+99981+99981 199991=9+99991+99991 199995=5+99995+99995 199999=1+99999+99999 1500000=500000+500000+500000 1818182=181818+818182+818182 1909091=90909+909091+909091 1950000=50000+950000+950000 1980199=19801+980199+980199 1995000=5000+995000+995000 1998002=1998+998002+998002 1999001=999+999001+999001 1999010=990+999010+999010 1999091=909+999091+999091 1999500=500+999500+999500 1999801=199+999801+999801 1999901=99+999901+999901 1999950=50+999950+999950 1999981=19+999981+999981 1999991=9+999991+999991 1999995=5+999995+999995 1999999=1+999999+999999 15000000=5000000+5000000+5000000 18181819=1818181+8181819+8181819 19500000=500000+9500000+9500000 19801981=198019+9801981+9801981 19900991=99009+9900991+9900991 19909091=90909+9909091+9909091 19950000=50000+9950000+9950000 19980020=19980+9980020+9980020 19990010=9990+9990010+9990010 19995000=5000+9995000+9995000 19998001=1999+9998001+9998001 19999001=999+9999001+9999001 19999010=990+9999010+9999010 19999091=909+9999091+9999091 19999500=500+9999500+9999500 19999801=199+9999801+9999801 19999901=99+9999901+9999901 19999950=50+9999950+9999950 19999981=19+9999981+9999981 19999991=9+9999991+9999991 19999995=5+9999995+9999995 19999999=1+9999999+9999999 150000000=50000000+50000000+50000000 181818182=18181818+81818182+81818182 190909091=9090909+90909091+90909091 195000000=5000000+95000000+95000000 198019802=1980198+98019802+98019802 199009901=990099+99009901+99009901 199500000=500000+99500000+99500000 199800200=199800+99800200+99800200 199900100=99900+99900100+99900100 199900991=99009+99900991+99900991 199909091=90909+99909091+99909091 199950000=50000+99950000+99950000 199980002=19998+99980002+99980002 199990001=9999+99990001+99990001 199990010=9990+99990010+99990010 199995000=5000+99995000+99995000 199998001=1999+99998001+99998001 199999001=999+99999001+99999001 199999010=990+99999010+99999010 199999091=909+99999091+99999091 199999500=500+99999500+99999500 199999801=199+99999801+99999801 199999901=99+99999901+99999901 199999950=50+99999950+99999950 199999981=19+99999981+99999981 199999991=9+99999991+99999991 199999995=5+99999995+99999995 199999999=1+99999999+99999999
On Mon, Feb 24, 2020 at 3:58 AM Éric Angelini <bk263401@skynet.be> wrote:
Hello Math-Fun, do you know any other solutions? (especially where A, B and C are distinct)
1818 = A 8182 = B + 8182 = C ------ = 18182 = D
Best, É.
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participants (5)
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Frank Stevenson -
Michael Greenwald -
Tom Duff -
Éric Angelini -
Éric Angelini