A349875 Triangular numbers whose mean digit value reaches a new maximum.

0, 1, 3, 6, 78, 686999778, 9876799878, 89996788896, 77779987999896, 589598998999878, 999699998689998991, 9988894989978899995, 95898999989999989765, 999999966989999986978996

Offset: 1

Author

Jon E. Schoenfield, Dec 03 2021

Keywords

• nonn
• base
• hard
• more

Comments

Subsequence of A068808.

No triangular number ends in 9, so the mean digit value is always less than 9.

Is this sequence finite? Or does the mean digit value approach some upper limit arbitrarily closely without ever reaching it exactly, and, if so, what is that limit?

a(14) <= 999999966989999986978996. - David A. Corneth, Dec 05 2021

Examples

			   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
		

Crossrefs

Cf. A000217, A068133, A068808, A069669, A069670, A095864.

Programs

Mathematica

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 *)

Python

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

Extensions

a(14) verified by Martin Ehrenstein, Dec 06 2021