A178053 -id:A178053 - cp's OEIS frontend

A134941 Mountain numbers.

1, 121, 131, 141, 151, 161, 171, 181, 191, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1321, 1341, 1351, 1361, 1371, 1381, 1391, 1421, 1431, 1451, 1461, 1471, 1481, 1491, 1521, 1531, 1541, 1561, 1571, 1581, 1591, 1621, 1631, 1641, 1651, 1671, 1681, 1691, 1721

Offset: 1

Omar E. Pol, Nov 22 2007

For n > 1 the structure of digits represents a mountain. The first digit is 1. The last digit is 1. The first digits are in increasing order. The last digits are in decreasing order. The numbers only have one largest digit. This sequence is finite. The last term is 12345678987654321.
The total number of terms is 21846. - Hans Havermann, Nov 25 2007
A002450(8) + 1 = 21846. - Reinhard Zumkeller, May 17 2010
From Reinhard Zumkeller, May 25 2010: (Start)
A178333 is the characteristic function of mountain numbers: A178333(a(n)) = 1;
A178334(n) is the number of mountain numbers <= n;
A178052 and A178053 give sums of digits and digital roots of mountain numbers;
A178051(n) is the peak value of the n-th mountain number. (End)

			The A-number of this sequence (A134941) is itself a mountain number:
  . . . 9 . .
  . . . . . .
  . . . . . .
  . . . . . .
  . . . . . .
  . . 4 . 4 .
  . 3 . . . .
  . . . . . .
  1 . . . . 1

Joshua Zucker, Table of n, a(n) for n = 1..21846 (shows all terms).

Cf. A134853, A135417, A134951, A182721.

Cf. A115300, A175044. - Reinhard Zumkeller, May 25 2010

import Data.List (elemIndices)
a134941 n = a134941_list !! (n-1)
a134941_list = elemIndices 1 a178333_list
-- Reinhard Zumkeller, Oct 28 2001

mountainQ[n_] := MatchQ[ IntegerDigits[n], {1, a___, b_, c___, 1} /; OrderedQ[{1, a, b}, Less] && OrderedQ[ Reverse[{b, c, 1}], Less]]; mountainQ[1] = True; Select[Range[2000], mountainQ] (* Jean-François Alcover, Jun 13 2012 *)
Prepend[Union @@ ((FromDigits@#&/@Flatten[Table[Join[(k=Prepend[#,1]&/@
Subsets[Range[2,#-1]])[[i]], {#}, (Reverse@# & /@k)[[j]]],
{i, 2^(# - 2)}, {j, 2^(# - 2)}], 1])&/@Range[9]), 1] (* Hans Rudolf Widmer, Apr 30 2024 *)

from itertools import product
def ups():
    d = "23456789"
    for b in product([0, 1], repeat=8):
        yield "1" + "".join(d[i]*b[i] for i in range(8))
def downsfrom(apex):
    if apex < 3: yield "1"*int(apex==2); return
    d = "8765432"[-(apex-2):]
    for b in product([0, 1], repeat=len(d)):
        yield "".join(d[i]*b[i] for i in range(len(d))) + "1"
def A134941(): # return full sequence as a list
    mountain_strs = (u+d for u in ups() for d in downsfrom(int(u[-1])))
    return sorted(int(ms) for ms in mountain_strs)
print(A134941()[:45]) # Michael S. Branicky, Dec 31 2021

A178052 Digit sums of mountain numbers (cf. A134941).

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 7, 9, 10, 11, 12, 13, 14, 8, 9, 11, 12, 13, 14, 15, 9, 10, 11, 13, 14, 15, 16, 10, 11, 12, 13, 15, 16, 17, 11, 12, 13, 14, 15, 17, 18, 12, 13, 14, 15, 16, 17, 19, 13, 14, 15, 16, 17, 18, 19, 9, 11, 12, 13, 14, 15, 16, 10, 11, 13, 14

Offset: 1

Reinhard Zumkeller, May 25 2010

a(n) = A007953(A134941(n));
a(n) can be interpreted as the volume of the "mountain" A134941(n);
a(21846) = A007953(12345678987654321) = A000217(9)+A000217(8) = 45+36 = 81 is the largest term, as A134941(21846)=12345678987654321 is the most voluminous "mountain".

			n=552, A134941(552)=134941 (example from A134941):
. . level 9: . . . . . . . . X . . . . . .
. . level 8: . . . . . . . . X . . . . . .
. . level 7: . . . . . . . . X . . . . . .
. . level 6: . . . . . . . . X . . . . . .
. . level 5: . . . . . . . . X . . . . . .
. . level 4: . . . . . . X . X . X . . . .
. . level 3: . . . . X . X . X . X . . . .
. . level 2: . . . . X . X . X . X . . . .
. . level 1: . . X . X . X . X . X . X . .
---------- --------------------------------
. . . . a(552) = 1 + 3 + 4 + 9 + 4 + 1 = 22.

R. Zumkeller, Table of n, a(n) for n = 1..21846 (full sequence)

Cf. A178053.

cp's OEIS Frontend

A134941 Mountain numbers.

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