

A096032


Take pairs (a, b), sorted on a, such that T(a)+T(b)=concatenation of a and b, where T(k) is the kth triangular number A000217(k). Sequence gives values of b.


6



1, 415, 1545, 1726, 2196, 910, 3676, 3846, 910, 5226, 415, 6970, 7171, 8526, 9231, 9300, 9756, 9850, 9880, 44835, 9880, 9850, 9756, 9300, 9231, 52830, 8526, 7171, 6970, 5226, 3846, 3676, 2196, 1726, 1545, 84906, 89386, 99580, 99580, 89386, 84906
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OFFSET

1,2


COMMENTS

It is easier to generate the pairs sorted by b. A ddigit number b is a member iff 4*(10^(2*d)10^db^2+b)+1 is a square. All such b occur twice, except for 1, which occurs once. There are no members with 2, 6, 7, or 8 digits. There are six distinct ninedigit members.  David Wasserman, May 15 2007


REFERENCES

J. S. Madachy, Madachy's Mathematical Recreations, pp. 166 Dover NY 1979.


LINKS



EXAMPLE

1726 of the sequence forms a pair with 150 and we indeed have T(150)+T(1726)=11325+1490401=1501726.


MATHEMATICA

f[n_] := Block[{k = n + 1, t1 = n(n + 1)/2, td = IntegerDigits[n]}, While[k < 15*n && t1 + k(k + 1)/2 != FromDigits[ Join[ td, IntegerDigits[k]]], k++ ]; If[k != 15*n, k, 0]]; Do[ k = f[n]; If[k != 0, Print[n, " & ", k]], {n, 10^6}] (* Robert G. Wilson v, Jun 21 2004 *)


CROSSREFS



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



