AUTHOR
_Amarnath Murthy (amarnath_murthy(AT)yahoo.com), _, Apr 18 2002
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_Amarnath Murthy (amarnath_murthy(AT)yahoo.com), _, Apr 18 2002
For k > 0, sequence contains (10^k+2)/2, (10^k+2)/3, (10^k+2)/6 and (10^(6k-1)+2)/7. The only other terms are 1 and 3. - _David Wasserman (wasserma(AT)spawar.navy.mil), _, Aug 25 2005
More terms from _David Wasserman (wasserma(AT)spawar.navy.mil), _, Aug 25 2005
Edited and extended by _Robert G. Wilson v (rgwv(AT)rgwv.com), _, Apr 22 2002
For k > 0, sequence contains (10^k+2)/2, (10^k+2)/3, (10^k+2)/6, and (10^(6k-1)+2)/7. The only other terms are 1 and 3. - David Wasserman (wasserma(AT)spawar.navy.mil), Aug 25 2005
base,nonn,new
1, 2, 3, 4, 6, 17, 34, 51, 167, 334, 501, 1667, 3334, 5001, 14286, 16667, 33334, 50001, 166667, 333334, 500001, 1666667, 3333334, 5000001, 16666667, 33333334, 50000001, 166666667, 333333334, 500000001, 1666666667, 3333333334
With the exception of the first term (1), each term in this sequence also evenly divides the number formed by concatenating (n-1) and (n-2). So, for example, 17 divides evenly into both 1819 and 1615. There do not appear to be any terms such that n divides evenly into the concatenation of (n-1) and (n-2) that are not included in this sequence. - Chuck Seggelin (barkeep(AT)plastereddragon.com), Oct 18 2003
For values of n with 3 digits or more, an apparent pattern arises such that the terms 1(6)...7, 3(3)...4, and 5(0)...1 are members of the sequence for each successive power of 10. The only exception to this in the terms given above is 14286. Cases like this only seem to arise when the term 16...7 is divisible by 7. Let x=16...7. If 7 divides x, then x/7*6 is a term. This works for 16667 since 16667 = 7*2381 and 14286 = 6*2381. The next n of the form 16...7 that is divisible by 7 is 16666666667 (7 * 2380952381), 16666666667/7*6 = 14285714286 and 14285714286 is a member of this sequence. 16666666666666667 is also divisible by 7 and 16666666666666667/7*6 = 14285714285714286 and 14285714285714286 is a member of this sequence. Note the similarity between terms of the form (142857)...14286 and the floating point representation of the fraction 1/7 (.142857142857142857...) - Chuck Seggelin (barkeep(AT)plastereddragon.com), Oct 18 2003
For k > 0, sequence contains (10^k+2)/2, (10^k+2)/3, (10^k+2)/6, and (10^(6k-1)+2)/7. The only other terms are 1 and 3. - David Wasserman (wasserma(AT)spawar.navy.mil), Aug 25 2005
base,nonn,new
More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Aug 25 2005
With the exception of the first term (1), each term in this sequence also evenly divides the number formed by concatenating (n-1) and (n-2). So, for example, 17 divides evenly into both 1819 and 1615. There do not appear to be any terms such that n divides evenly into the concatenation of (n-1) and (n-2) that are not included in this sequence. - Chuck Seggelin (barkeep(AT)plastereddragon.com), Oct 18 2003
For values of n with 3 digits or more, an apparent pattern arises such that the terms 1(6)...7, 3(3)...4, and 5(0)...1 are members of the sequence for each successive power of 10. The only exception to this in the terms given above is 14286. Cases like this only seem to arise when the term 16...7 is divisible by 7. Let x=16...7. If 7 divides x, then x/7*6 is a term. This works for 16667 since 16667 = 7*2381 and 14286 = 6*2381. The next n of the form 16...7 that is divisible by 7 is 16666666667 (7 * 2380952381), 16666666667/7*6 = 14285714286 and 14285714286 is a member of this sequence. 16666666666666667 is also divisible by 7 and 16666666666666667/7*6 = 14285714285714286 and 14285714285714286 is a member of this sequence. Note the similarity between terms of the form (142857)...14286 and the floating point representation of the fraction 1/7 (.142857142857142857...) - Chuck Seggelin (barkeep(AT)plastereddragon.com), Oct 18 2003
Select[ Range[10^7], IntegerQMod[ ToExpressionFromDigits[ StringJoinJoin[ ToStringIntegerDigits[ # + 1], ToStringIntegerDigits[ # + 2]]] / , # ] == 0 & ]
base,nonn,new
Numbers n that divide the concatenation of n+1 and n+2.
1, 2, 3, 4, 6, 17, 34, 51, 167, 334, 501, 1667, 3334, 5001, 14286, 16667, 33334, 50001, 166667, 333334, 500001, 1666667, 3333334, 5000001
1,2
17 divides 1819 hence is a member.
Select[ Range[10^7], IntegerQ[ ToExpression[ StringJoin[ ToString[ # + 1], ToString[ # + 2]]] / # ] & ]
base,nonn
Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 18 2002
Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 22 2002
approved