A357172 -id:A357172 - 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

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.

Original entry on oeis.org

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

A358100 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly decreasing order.

Original entry on oeis.org

1, 2, 4, 6, 12, 20, 30, 40, 80, 60, 252, 120, 240, 540, 360, 630, 420, 960, 1440, 840, 1260, 2880, 3360, 4320, 2520, 6720, 5040, 8640, 10080, 15120, 50400, 20160, 40320, 30240, 171360, 90720, 383040, 60480, 120960, 181440, 362880, 544320, 937440, 786240, 2056320

Offset: 1

Bernard Schott, Nov 01 2022

This sequence is finite since A009995 is finite with 1022 nonzero terms, hence the last term is a(1022) = lcm of the 1022 positive terms of A009995.

			For n=7, the divisors of 30 are {1, 2, 3, 5, 6, 10, 15, 30} of which 7 have their decimal digits in strictly decreasing order (all except 15). No integer < 30 has 7 such divisors, so a(7) = 30.

Cf. A009995, A190219, A358099, A358101.

Similar: A087997 (palindromic), A355303 (undulating), A357172 (increasing order).

s[n_] := DivisorSum[n, 1 &, Greater @@ 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[45, 3*10^6] (* Amiram Eldar, Nov 01 2022 *)

f(n) = sumdiv(n, d, my(dd=digits(d)); vecsort(dd, , 12) == dd); \\ A358099
a(n) = my(k=1); while(f(k)!=n, k++); k; \\ Michel Marcus, Nov 01 2022

More terms from Amiram Eldar, Nov 01 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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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.

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A358100 a(n) is the smallest integer that has exactly n divisors whose decimal digits are in strictly decreasing order.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions