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
Examples
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.
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)
Crossrefs
Programs
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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
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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 *) -
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
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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
Formula
a(n) >= A005179(n). - David A. Corneth, Jan 25 2023
Extensions
More terms from David A. Corneth, Jul 08 2022
Comments