cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A134941 Mountain numbers.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Nov 22 2007

Keywords

Comments

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)

Examples

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

Links

Crossrefs

Cf. A134853, A135417, A134951, A182721.
Cf. A115300, A175044. - Reinhard Zumkeller, May 25 2010

Programs

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

A178334 Number of mountain numbers <= n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Reinhard Zumkeller, May 25 2010

Keywords

Comments

a(n) = 21846 for n >= 12345678987654321.

Links

Crossrefs

Cf. A135417.

Programs

  • Python
    from itertools import count, islice
def agen(): # generator of terms; uses code in A134941
    A134941_full = A134941() + [-1]
    c = i = 0
    for j in count(0):
        if j == A134941_full[i]: i, c = i+1, c+1
        yield c
print(list(islice(agen(), 122))) # Michael S. Branicky, Jan 09 2023

Formula

a(n) = Sum_{k=0..n} A178333(k).

A182721 Mountain emirps.

Original entry on oeis.org

1231, 1321, 1381, 1471, 1741, 1831, 12491, 12641, 12841, 13591, 13751, 13781, 13841, 14591, 14621, 14821, 14831, 14891, 15731, 15791, 18731, 19421, 19531, 19541, 19751, 19841, 123731, 123821, 124951, 124981, 125641, 125651, 125791, 125821, 125941, 126761, 126851
Offset: 1

Views

Author

Omar E. Pol, Dec 21 2010

Keywords

Comments

Intersection of emirps A006567 and mountain numbers A134941.
The smallest mountain emirp 1231 and other terms of this sequence was mentioned by Loungrides in Prime Curios! (see link).
Question: How many are there?
There are 602 such terms. - Michael S. Branicky, Dec 31 2021

Examples

			Illustration of a(11) = 13751 as a mountain emirp:
  . . . . .
  . . . . .
  . . 7 . .
  . . . . .
  . . . 5 .
  . . . . .
  . 3 . . .
  . . . . .
  1 . . . 1

Links

Crossrefs

Cf. A006567, A134941, A134951, A135417, A173071.

Programs

  • Python
    # uses A134941()
from sympy import isprime
def is_emirp(n):
    if not isprime(n): return False
    revn = int(str(n)[::-1])
    return n != revn and isprime(revn)
print([k for k in A134941() if is_emirp(k)]) # Michael S. Branicky, Dec 31 2021

Formula

A006567 INTERSECT A134941.

Extensions

More terms from Nathaniel Johnston, Dec 29 2010
Terms a(31) and beyond from Michael S. Branicky, Dec 31 2021

A178912 Number of generalized mountain numbers (see A134853) with n digits.

Original entry on oeis.org

9, 0, 240, 1380, 4578, 10794, 19494, 27912, 32195, 30085, 22748, 13820, 6656, 2486, 695, 137, 17, 1
Offset: 1

Views

Author

Nathaniel Johnston, Dec 29 2010

Keywords

Comments

The total number of generalized mountain numbers is 173247.

Examples

			a(18) = 1 because there is exactly one generalized mountain number with 18 digits: 123456789876543210

Crossrefs

Cf. A134853, A134941, A135417
Showing 1-4 of 4 results.

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