A355593 -id:A355593 - cp's OEIS frontend

A355595 Positions of records in A355593: Integers whose number of alternating divisors sets a new record.

1, 2, 4, 6, 12, 24, 36, 72, 144, 180, 360, 504, 630, 1080, 1260, 1890, 2520, 3780, 7560, 15120, 18900, 22680, 30240, 37800, 45360, 75600, 90720, 151200, 162540, 226800, 317520, 325080, 650160, 763560, 1137780, 1243620, 1527120, 2275560, 3054240, 3738420, 4551120, 6826680, 7476840, 14953680, 17445960, 21818160, 26168940, 36363600, 43636320, 52337880

Offset: 1

Bernard Schott, Jul 08 2022

Alternating integers are in A030141.

Corresponding records of number of alternating divisors are 1, 2, 3, 4, 6, 7, 9, 11, ...

			a(6) = 24 is in the sequence because A355593(24) = 7 is larger than any earlier value in A355593.

Cf. A030141, A355593, A355594.

Similar, but with undulating divisors: A355304.

q[n_] := ! MemberQ[Differences[Mod[IntegerDigits[n], 2]], 0]; f[n_] := DivisorSum[n, 1 &, q[#] &]; fm = 0; s = {}; Do[fn = f[n]; If[fn > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 08 2022 *)

a(21)-a(36) from Amiram Eldar, Jul 08 2022

a(37)-a(50) from Charles R Greathouse IV, Jul 08 2022

A355594 a(n) is the smallest integer that has exactly n alternating divisors.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 24, 48, 36, 96, 72, 144, 210, 180, 420, 360, 504, 864, 630, 1080, 1512, 2160, 1260, 3150, 1890, 2520, 5040, 6300, 3780, 10080, 12600, 9450, 7560, 32760, 15120, 18900, 22680, 30240, 88830, 37800, 45360, 75600, 105840, 90720, 151200, 162540, 254520

Offset: 1

Bernard Schott, Jul 08 2022

This sequence first differs from A005179 at index 7 where A005179(7) = 64.

			16 has 5 divisors: {1, 2, 4, 8, 16} all of which are alternating integers; no positive integer smaller than 16 has five alternating divisors, hence a(5) = 16.
96 has 12 divisors: {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}, only 24 and 48 are not alternating; no positive integer smaller than 96 has ten alternating divisors, hence a(10) = 96.

David A. Corneth, Table of n, a(n) for n = 1..147 (first 107 terms from Robert Israel)
David A. Corneth, Some upper bounds on a(n)

Cf. A005179, A030141 (alternating numbers), A355593, A355595, A355596.

Similar, but with undulating divisors: A355303.

isalt:= proc(n) local L; option remember;
   L:= convert(n,base,10) mod 2;
   L:= L[2..-1]-L[1..-2];
   not member(0,L)
end proc:
N:= 50: # for a(1)..a(N)
V:= Vector(N): count:= 0:
for n from 1 while count < N do
  w:= nops(select(isalt,numtheory:-divisors(n)));
  if w <= N and V[w] = 0 then V[w]:= n; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Jan 24 2023

q[n_] := ! MemberQ[Differences[Mod[IntegerDigits[n], 2]], 0]; f[n_] := DivisorSum[n, 1 &, q[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^6] (* Amiram Eldar, Jul 08 2022 *)

is(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1; \\ A030141
a(n) = my(k=1); while (sumdiv(k, d, is(d)) != n, k++); k; \\ Michel Marcus, Jul 11 2022

from itertools import count
from sympy import divisors
def A355594(n):
    for m in count(1):
        if sum(1 for k in divisors(m,generator=True) if all(int(a)+int(b)&1 for a, b in zip(str(k),str(k)[1:]))) == n:
            return m # Chai Wah Wu, Jul 12 2022

a(n) >= A005179(n). - David A. Corneth, Jan 25 2023

More terms from David A. Corneth, Jul 08 2022

A355596 Numbers all of whose divisors are alternating numbers (A030141).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 32, 36, 41, 43, 47, 49, 50, 54, 58, 61, 63, 67, 69, 81, 83, 87, 89, 94, 98, 101, 103, 107, 109, 123, 125, 127, 129, 141, 145, 147, 149, 161, 163, 167, 181, 183, 189, 214, 218, 250, 254, 290, 298

Offset: 1

Bernard Schott, Jul 12 2022

The smallest alternating number that is not a term is 30, because of 15.

			32 is a term since all the divisors of 32, i.e., 1, 2, 4, 8, 16 and 32, are alternating numbers

Subsequence of A030141.
Cf. A355593, A355594, A355595.
Similar sequences: A062687, A190217, A329419, A337941.

q[n_] := AllTrue[Divisors[n], !MemberQ[Differences[Mod[IntegerDigits[#], 2]], 0] &]; Select[Range[300], q] (* Amiram Eldar, Jul 12 2022 *)

isokd(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1; \\ A030141
isok(m) = sumdiv(m, d, isokd(d)) == numdiv(m); \\ Michel Marcus, Jul 12 2022

from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
    if n < 10: return True
    s = str(n)
    return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-1))
def ok(n):
    return c(n) and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(1, 200) if ok(k)]) # Michael S. Branicky, Jul 12 2022

a(51) and beyond from Michael S. Branicky, Jul 12 2022

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

Original entry on oeis.org

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

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

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A355595 Positions of records in A355593: Integers whose number of alternating divisors sets a new record.

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A355594 a(n) is the smallest integer that has exactly n alternating divisors.

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A355596 Numbers all of whose divisors are alternating numbers (A030141).

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

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

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