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

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

1,2

Comments

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

Links

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)

Formula

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(PARI) 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
(Python)
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

Crossrefs

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

Similar, but with undulating divisors: A355303.

Sequence in context: A209867 A136033 A343019 * A357172 A355303 A099315

Adjacent sequences: A355591 A355592 A355593 * A355595 A355596 A355597

Keyword

nonn,base

Author

Bernard Schott, Jul 08 2022

Extensions

More terms from David A. Corneth, Jul 08 2022

Status

approved