A099150
Positive integers k such that f(k)+f(k)=concatenation of k and k, where f(k)=k(k+3)/2 (A000096).
Original entry on oeis.org
8, 98, 998, 9998, 99998, 999998, 9999998, 99999998, 999999998, 9999999998, 99999999998, 999999999998, 9999999999998, 99999999999998, 999999999999998, 9999999999999998, 99999999999999998, 999999999999999998, 9999999999999999998, 99999999999999999998
Offset: 1
99998*(99998+3) = 9999899998 (concatenation of 99998 and 99998).
A099148
Take a <= b such that f(a)+f(b)=concatenation of a and b, where f(k)=k(k+3)/2 (A000096). Sequence gives values of a.
Original entry on oeis.org
2, 3, 8, 9, 14, 15, 20, 39, 77, 98, 99, 138, 158, 164, 177, 200, 399, 492, 614, 998, 999, 1065, 1383, 1505, 1598, 1797, 1859, 1920, 2000, 2655, 3077, 3213, 3762, 3999, 6707
Offset: 1
A099149
Take a <= b such that f(a)+f(b)=concatenation of a and b, where f(k)=k(k+3)/2 (A000096). Sequence gives values of b.
Original entry on oeis.org
5, 6, 8, 8, 6, 5, 59, 78, 384, 98, 98, 506, 78, 1803, 59, 599, 798, 960, 921, 998, 998, 14555, 921, 860, 798, 599, 506, 384, 5999, 6785, 24615, 7343, 27170, 7998, 36005
Offset: 1
From _R. J. Mathar_, Oct 10 2017 (Start)
Values of a, then b, then f(a)+f(b) and concatenations (first a, then b or first b, then a), sorted on b, are:
0 0 0+0=0
2 5 5+20=25
3 6 9+27=36
8 8 44+44=88
8 9 44+54=98
6 14 27+119=146
5 15 20+135=155
0 17 0+170=170
20 59 230+1829=2059
39 78 819+3159=3978
98 98 4949+4949=9898
98 99 4949+5049=9998
78 158 3159+12719=15878
59 177 1829+15930=17759
77 384 3080+74304=77384
138 506 9729+128777=138506
200 599 20300+180299=200599
399 798 80199+319599=399798
492 860 121770+371090=492860
614 921 189419+425502=614921
998 998 499499+499499=998998
998 999 499499+500499=999998
921 1383 425502+958419=1383921
860 1505 371090+1134770=1505860
798 1598 319599+1279199=1598798
599 1797 180299+1617300=1797599
164 1803 13694+1628109=1641803
506 1859 128777+1730729=1859506
384 1920 74304+1846080=1920384
2000 5999 2003000+18002999=20005999
2655 6785 3528495+23028290=26556785
3213 7343 5166504+26970839=32137343
3999 7998 8001999+31995999=39997998
(End)
A096031
Take pairs (a, b), sorted on a, such that T(a)+T(b)=concatenation of a and b, where T(k) is the k-th triangular number A000217(k). Sequence gives values of a.
Original entry on oeis.org
19, 90, 120, 150, 244, 585, 700, 769, 1414, 1474, 1909, 2829, 3030, 4774, 6154, 6324, 7804, 8274, 8455, 10614, 11544, 11725, 12195, 13675, 13845, 15094, 15225, 16969, 17170, 18525, 19230, 19299, 19755, 19849, 19879, 47170, 55165, 90844, 109155
Offset: 1
244 of the sequence forms a pair with 2196 and we indeed have T(244)+T(2196)=29890+2412306=2442196.
- J. S. Madachy, Madachy's Mathematical Recreations, pp. 166 Dover NY 1979.
-
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 *)
A099151
Positive integers a such that f(3a)+f(a)=concatenation of 3a and a, where f(k)=k(k+3)/2 (A000096).
Original entry on oeis.org
5, 59, 599, 5999, 59999, 599999, 5999999, 59999999, 599999999, 5999999999, 59999999999, 599999999999, 5999999999999, 59999999999999, 599999999999999, 5999999999999999, 59999999999999999, 599999999999999999, 5999999999999999999, 59999999999999999999, 599999999999999999999
Offset: 1
599 is in the sequence because (3*599)(3*599+3)/2 + 599(602)/2 = 1797*1800/2 + 599*602/2 = 1797599.
Showing 1-5 of 5 results.
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