A357173 -id:A357173 - cp's OEIS frontend

A357171 a(n) is the number of divisors of n whose digits are in strictly increasing order (A009993).

1, 2, 2, 3, 2, 4, 2, 4, 3, 3, 1, 6, 2, 4, 4, 5, 2, 6, 2, 4, 3, 2, 2, 8, 3, 4, 4, 6, 2, 6, 1, 5, 2, 4, 4, 9, 2, 4, 4, 5, 1, 6, 1, 3, 6, 4, 2, 10, 3, 4, 3, 5, 1, 7, 2, 8, 4, 4, 2, 8, 1, 2, 4, 5, 3, 4, 2, 6, 4, 6, 1, 11, 1, 3, 5, 5, 2, 8, 2, 6, 4, 2, 1, 9, 3, 2, 3, 4, 2, 9, 3, 5, 2, 3, 3, 10, 1, 5, 3, 5

Offset: 1

Bernard Schott, Sep 16 2022

As A009993 is finite with 512 terms, a(n) is bounded with a(n) <= 511 and not 512, since A009993(1) = 0.

			22 has 4 divisors {1, 2, 11, 22} of which two have decimal digits that are not in strictly increasing order: {11, 22}, hence a(22) = 4-2 = 2.
52 has divisors {1, 2, 4, 13, 26, 52} and a(52) = 5 of them have decimal digits that are in strictly increasing order (all except 52 itself).

Cf. A009993, A357172, A357173, A160218.

Similar: A087990 (palindromic), A355302 (undulating), A355593 (alternating).

f:= proc(n) local d,L,i,t;
  t:= 0;
  for d in numtheory:-divisors(n) do
    L:= convert(d,base,10);
    if `and`(seq(L[i]>L[i+1],i=1..nops(L)-1)) then t:= t+1 fi
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Sep 16 2022

a[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; Array[a, 100] (* Amiram Eldar, Sep 16 2022 *)

isok(d) = Set(d=digits(d)) == d; \\ A009993
a(n) = sumdiv(n, d, isok(d)); \\ Michel Marcus, Sep 16 2022

from sympy import divisors
def c(n): s = str(n); return s == "".join(sorted(set(s)))
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Sep 16 2022

G.f.: Sum_{n in A009993} x^n/(1-x^n). - Robert Israel, Sep 16 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n=2..512} 1/A009993(n) = 4.47614714667538759358... (this is a rational number whose numerator and denominator have 1037 and 1036 digits, respectively). - Amiram Eldar, Jan 06 2024

A357172 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly increasing order.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 54, 24, 36, 48, 72, 180, 144, 360, 336, 468, 504, 936, 1008, 1512, 2520, 3024, 5040, 6552, 7560, 22680, 13104, 19656, 49140, 105840, 39312, 78624, 98280, 248976, 334152, 196560, 393120, 668304, 1244880, 1670760, 1867320, 4520880, 3341520, 3734640

Offset: 1

Bernard Schott, Sep 16 2022

This sequence is finite since A009993 is finite with 511 nonzero terms, hence the last term is a(511) = lcm of the 511 nonzero terms of A009993.

a(511) = 8222356410...6120992000 and has 1036 digits. - Michael S. Branicky, Sep 16 2022

			For n=7, the divisors of 54 are {1, 2, 3, 6, 9, 18, 27, 54} of which 7 have their digits in strictly increasing order (all except 54). No integer < 54 has 7 such divisors, so a(7) = 54.

Cf. A009993, A357171, A357173, A160218.

Similar: A087997 (palindromic), A355303 (undulating), A355594 (alternating).

s[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; seq[len_, nmax_] := Module[{v = Table[0, {len}], n = 1, c = 0, i}, While[c < len && n < nmax, i = s[n]; If[i <= len && v[[i]] == 0, v[[i]] = n; c++]; n++]; v]; seq[25, 10^4] (* Amiram Eldar, Sep 16 2022 *)

isok(d) = Set(d=digits(d)) == d; \\ A009993
f(n) = sumdiv(n, d, isok(d)); \\ A357171
a(n) = my(k=1); while (f(k) !=n, k++); k; \\ Michel Marcus, Sep 16 2022

from sympy import divisors
from itertools import count, islice
def c(n): s = str(n); return s == "".join(sorted(set(s)))
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 37))) # Michael S. Branicky, Sep 16 2022

More terms from Amiram Eldar, Sep 16 2022

A358101 Positions of records in A358099, i.e., integers whose number of divisors whose decimal digits are in strictly decreasing order sets a new record.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 30, 40, 60, 120, 240, 360, 420, 840, 1260, 2520, 5040, 8640, 10080, 15120, 20160, 30240, 60480, 120960, 181440, 362880, 544320, 786240, 1572480, 1874880, 3749760, 5624640, 7862400, 14938560, 23587200, 24373440, 31872960, 63745920, 95618880

Offset: 1

Bernard Schott, Nov 03 2022

As A009995 is finite, this sequence is necessarily finite.

Corresponding records are 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, ...

			a(9) = 60 is in the sequence because A358099(60) = 10 is larger than any earlier value in A358099.

Cf. A009995, A190219, A358099, A358100.

Similar sequences: A093036, A340548, A357173.

f[n_] := DivisorSum[n, 1 &, Greater @@ IntegerDigits[#] &]; fm = 0; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Nov 03 2022 *)

More terms from Amiram Eldar, Nov 03 2022

cp's OEIS Frontend

A357171 a(n) is the number of divisors of n whose digits are in strictly increasing order (A009993).

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A357172 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly increasing order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A358101 Positions of records in A358099, i.e., integers whose number of divisors whose decimal digits are in strictly decreasing order sets a new record.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions