A357171 -id:A357171 - cp's OEIS frontend

A357173 Positions of records in A357171, i.e., integers whose number of divisors whose decimal digits are in strictly increasing order sets a new record.

1, 2, 4, 6, 12, 24, 36, 48, 72, 144, 336, 468, 504, 936, 1008, 1512, 2520, 3024, 5040, 6552, 7560, 13104, 19656, 39312, 78624, 98280, 196560, 393120, 668304, 1244880, 1670760, 1867320, 3341520, 3734640, 7469280, 22407840, 26142480, 31744440, 52284960, 63488880

Offset: 1

Bernard Schott, Sep 17 2022

As A009993 is finite, this sequence is necessarily finite.

Corresponding records are 1, 2, 3, 4, 6, 8, 9, 10, 11, ...

			a(6) = 24 is in the sequence because A357171(24) = 8 is larger than any earlier value in A357171.

Cf. A009993, A357171, A357172, A160218.

Similar sequences: A093036, A340548, A355595.

s[n_] := DivisorSum[n, 1 &, Less @@ IntegerDigits[#] &]; seq = {}; sm = 0; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 1, 10^4}]; seq (* Amiram Eldar, Sep 17 2022 *)

isok(d) = Set(d=digits(d)) == d; \\ A009993
f(n) = sumdiv(n, d, isok(d)); \\ A357171
lista(nn) = my(r=0, list = List()); for (k=1, nn, my(m=f(k)); if (m>r, listput(list, k); r = m);); Vec(list); \\ Michel Marcus, Sep 18 2022

More terms from Amiram Eldar, Sep 17 2022

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

A358099 a(n) is the number of divisors of n whose digits are in strictly decreasing order (A009995).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Oct 29 2022

As A009995 is finite with 1023 terms, a(n) is bounded with a(n) <= 1022 and not 1023, since A009995(1) = 0.

			22 has 4 divisors {1, 2, 11, 22} of which two have decimal digits that are not in strictly decreasing order: {11, 22}, hence a(22) = 4-2 = 2.
52 has 6 divisors {1, 2, 4, 13, 26, 52} of which four have decimal digits that are in strictly decreasing order {1, 2, 4, 52}, hence a(52) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A009995, A190219, A358100, A358101.

Similar: A086971 (semiprimes), A087990 (palindromic), A355593 (alternating), A357171 (increasing order).

f:= proc(n) local L;
   if n < 10 then return true fi;
   L:= convert(n,base,10);
   andmap(type,L[2..-1]-L[1..-2],positive)
end proc:
g:= n -> nops(select(f,numtheory:-divisors(n))):
map(g, [$1..100]); # Robert Israel, Oct 31 2022

a[n_] := DivisorSum[n, 1 &, Max @ Differences @ IntegerDigits[#] < 0 &]; Array[a, 100] (* Amiram Eldar, Oct 29 2022 *)

a(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, ,12) == dd); \\ Michel Marcus, Oct 30 2022

from sympy import divisors
def c(n): s = str(n); return all(s[i+1] < s[i] for i in range(len(s)-1))
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, Feb 12 2024

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n=2..1023} 1/A009995(n) = 3.89840673699905364734... (this is a rational number whose numerator and denominator have 1292 and 1291 digits, respectively). - Amiram Eldar, Jan 06 2024

cp's OEIS Frontend

A357173 Positions of records in A357171, i.e., integers whose number of divisors whose decimal digits are in strictly increasing order sets a new record.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

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

A358099 a(n) is the number of divisors of n whose digits are in strictly decreasing order (A009995).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula