A355594 - OEIS

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

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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.