A068092
Index of smallest triangular number with n digits.
Original entry on oeis.org
1, 4, 14, 45, 141, 447, 1414, 4472, 14142, 44721, 141421, 447214, 1414214, 4472136, 14142136, 44721360, 141421356, 447213595, 1414213562, 4472135955, 14142135624, 44721359550, 141421356237, 447213595500, 1414213562373, 4472135955000, 14142135623731
Offset: 1
a(4) = 45 as the 45th triangular number is 45*46/2 = 1035 while the 44th is 990.
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[Round(Sqrt(2*10^(n-1))) : n in [1..30]]; // Vincenzo Librandi, Oct 05 2011
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f[n_] := Block[{a = Floor[Sqrt[2*10^n]]}, If[a(a + 1)/2 < 10^n, a++ ]; Return[a]]; Table[ f[n], {n, 0, 30} ]
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a(n) = round(sqrt(2*10^(n-1))) \\ Charles R Greathouse IV, Oct 04 2011
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from math import isqrt
def A068092(n): return isqrt(10**(n-1)<<3)+1>>1 # Chai Wah Wu, Oct 17 2022
A095866
Largest n-digit number - largest n-digit triangular number.
Original entry on oeis.org
3, 8, 9, 129, 318, 1008, 2843, 8988, 38439, 121089, 42708, 88208, 2034819, 1749819, 2235879, 104271309, 447194784, 1234742858, 2234186964, 3266133123, 22172578524, 114481278033, 204796672749, 841526085621, 355055477499
Offset: 1
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lndt[n_]:=Module[{c=Floor[(Sqrt[8*10^n+1]-1)/2]},(c(c+1))/2]; Join[{3}, Table[ 10^n-1-lndt[n],{n,2,30}]] (* Harvey P. Dale, May 16 2017 *)
A068093
Smallest n-digit triangular number.
Original entry on oeis.org
1, 10, 105, 1035, 10011, 100128, 1000405, 10001628, 100005153, 1000006281, 10000020331, 100000404505, 1000001326005, 10000002437316, 100000012392316, 1000000042485480, 10000000037150046, 100000000000018810, 1000000000179470703, 10000000002237948990
Offset: 1
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triInverse[n_] := Floor[(Sqrt[1 + 8*n] - 1)/2]; tri[n_] := n*(n+1)/2; Table[tri[1 + triInverse[10^(n-1) - 1]], {n, 20}] (* T. D. Noe, Jul 27 2012 *)
A068094
Number of n-digit triangular numbers.
Original entry on oeis.org
3, 10, 31, 96, 306, 967, 3058, 9670, 30579, 96700, 305793, 967000, 3057922, 9670000, 30579224, 96699996, 305792239, 966999967, 3057922393, 9669999669, 30579223926, 96699996687, 305792239263, 966999966873, 3057922392627, 9669999668731, 30579223926265
Offset: 1
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triInverse[n_] := Floor[(Sqrt[1 + 8*n] - 1)/2]; Differences[Table[triInverse[10^(n - 1) - 1], {n, 30}]] (* T. D. Noe, Jul 27 2012 *)
A095863
Index of largest triangular number with n digits.
Original entry on oeis.org
3, 13, 44, 140, 446, 1413, 4471, 14141, 44720, 141420, 447213, 1414213, 4472135, 14142135, 44721359, 141421355, 447213594, 1414213561, 4472135954, 14142135623, 44721359549, 141421356236, 447213595499, 1414213562372
Offset: 1
A095865
Smallest n-digit triangular number - smallest n-digit number.
Original entry on oeis.org
0, 0, 5, 35, 11, 128, 405, 1628, 5153, 6281, 20331, 404505, 1326005, 2437316, 12392316, 42485480, 37150046, 18810, 179470703, 2237948990, 10876002500, 22548781025, 26940078203, 242416922750, 572687476751, 4117080477500
Offset: 1
A349875
Triangular numbers whose mean digit value reaches a new maximum.
Original entry on oeis.org
0, 1, 3, 6, 78, 686999778, 9876799878, 89996788896, 77779987999896, 589598998999878, 999699998689998991, 9988894989978899995, 95898999989999989765, 999999966989999986978996
Offset: 1
n a(n) digit sum #dgts mean digit value
-- -------------------- --------- ----- ----------------
1 0 0 1 0
2 1 1 1 1
3 3 3 1 3
4 6 6 1 6
5 78 15 2 7.5
6 686999778 69 9 7.66666666666...
7 9876799878 78 10 7.8
8 89996788896 87 11 7.90909090909...
9 77779987999896 111 14 7.92857142857...
10 589598998999878 120 15 8
11 999699998689998991 145 18 8.05555555555...
12 9988894989978899995 154 19 8.10526315789...
13 95898999989999989765 163 20 8.15
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seq = {}; max = -1; Do[If[(m = Mean @ IntegerDigits[(t = n*(n + 1)/2)]) > max, max = m; AppendTo[seq, t]], {n, 0, 10^6}]; seq (* Amiram Eldar, Dec 03 2021 *)
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def meandigval(n): s = str(n); return sum(map(int, s))/len(s)
def afind(limit):
alst, k, t, record = [], 0, 0, -1
while t <= limit:
mdv = meandigval(t)
if mdv > record:
print(t, end=", ")
record = mdv
k += 1
t += k
afind(10**14) # Michael S. Branicky, Dec 03 2021
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