A115300 -id:A115300 - 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

A257297 a(n) = (initial digit of n) * (n with initial digit removed).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 0, 1, 2, 3

Offset: 0

M. F. Hasler, May 10 2015

The initial 100 terms match those of A035930 (maximal product of any two numbers whose concatenation is n) and also those of A171765 (product of digits of n, or 0 for n<10), and except for initial terms, also A007954 (product of decimal digits of n) and A115300 (greatest digit of n * least digit of n).
Iterations of this map always end in 0, since a(n) < n. Sequence A257299 lists numbers for which these iterations reach 0 in exactly 9 steps, with the additional constraint of having each time a different initial digit.
If "initial" is replaced by "last" in the definition (A257850), then we get the same values up to a(100), but (10, 20, 30, ...) for n = 101, 102, 103, ..., again different from each of the 4 other sequences mentioned in the first comment. - M. F. Hasler, Sep 01 2021

			For n<10, a(n) = n*0 = 0, since removing the initial and only digit leaves nothing, i.e., zero (by convention).
a(10) = 1*0 = 0, a(12) = 1*2 = 2, ..., a(20) = 2*0 = 0, a(21) = 2*1 = 2, a(22) = 2*2 = 4, ...
a(99) = 9*9 = 81, a(100) = 1*00 = 0, a(101) = 1*01 = 1, ..., a(123) = 1*23, ...

Cf. A257850, A007954, A035930, A080464, A115300, A169669, A111707, A088117, A187844.

a:= n-> `if`(n<10, 0, (s-> parse(s[1])*parse(s[2..-1]))(""||n)):
seq(a(n), n=0..120);  # Alois P. Heinz, Feb 12 2024

Table[Times@@FromDigits/@TakeDrop[IntegerDigits@n,1],{n,0,103}] (* Giorgos Kalogeropoulos, Sep 03 2021 *)

apply( {A257297(n)=vecprod(divrem(n,10^logint(n+!n,10)))}, [0..111]) \\ Edited by M. F. Hasler, Sep 01 2021

def a(n): s = str(n); return 0 if len(s) < 2 else int(s[0])*int(s[1:])
print([a(n) for n in range(104)]) # Michael S. Branicky, Sep 01 2021

For 1 <= m <= 9 and n < 10^k, a(m*10^k + n) = m*n.

a(101..103) corrected by M. F. Hasler, Sep 01 2021

A169669 (first digit of n) * (last digit of n) in decimal representation.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0

Offset: 0

Reinhard Zumkeller, Apr 05 2010

a(n) = A000030(n)*A010879(n);
a(n) = A115300(n) for n<=100, A115300(101) = 0;
a(n) = A111707(n) for n<=109, A111707(110) = 1;
0 <= a(n) <= 81, range = A174995;
a(10*n + n mod 10) = a(n);
a(A008592(n)) = 0;
A000030(a(A017281(n))) = A000030(A017281(n));
a(n) = a(A004086(n))*A168184(n);

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A088133.

Cf. A000030, A008592, A010879, A111707, A115300, A174995.

a169669 n = a000030 n * mod n 10
-- Reinhard Zumkeller, Apr 29 2015

def a(n): return int(str(n)[0])*(n%10)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 13 2022

A115299 Greatest digit of n + least digit of n. Different from A088133.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 8, 9, 10, 11, 12, 13, 14

Offset: 1

Rick L. Shepherd, Jan 20 2006

a(101) = 1 and A088133(101) = 2, but all previous terms match.

			a(1) = 1 + 1 = 2, a(232) = 3 + 2 = 5, a(1889009898) = 9 + 0 = 9.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A037904 (greatest-least), A115300 (greatest*least), A088133 (first+last).

Array[Max[#] + Min[#] &@ IntegerDigits[#] &, 120] (* Michael De Vlieger, Dec 12 2023 *)

def a(n): d = list(map(int, str(n))); return max(d) + min(d)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Dec 12 2023

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A134941 Mountain numbers.

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A257297 a(n) = (initial digit of n) * (n with initial digit removed).

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A169669 (first digit of n) * (last digit of n) in decimal representation.

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A115299 Greatest digit of n + least digit of n. Different from A088133.

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