A356019 a(n) is the smallest number that has exactly n evil divisors (A001969).
1, 3, 6, 12, 18, 45, 30, 135, 72, 60, 90, 765, 120, 1575, 270, 180, 600, 3465, 480, 13545, 360, 540, 1530, 10395, 1260, 720, 3150, 1980, 1080, 49725, 1440, 45045, 2520, 3060, 6930, 2160, 3780, 58905, 27090, 6300, 5040, 184275, 4320, 135135, 6120, 7920, 20790, 329175, 7560, 8640
Offset: 0
Examples
a(4) = 18 since 18 has six divisors: {1, 2, 3, 6, 9, 18} of which four {3, 6, 9, 18} have an even number of 1's in their binary expansion: 11, 110, 1001 and 10010 respectively; also, no positive integer smaller than 18 has exactly four divisors that are evil.
Links
- David A. Corneth, Table of n, a(n) for n = 0..396
- David A. Corneth, Upperbounds on a(n), terms <= 8*10^9 are certain
Crossrefs
Programs
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Maple
# output in unsorted b-file style A356019_list := [seq(0,i=1..1000)] ; for n from 1 do evd := A356018(n) ; if evd < nops(A356019_list) then if op(evd+1,A356019_list) <= 0 then printf("%d %d\n",evd,n) ; A356019_list := subsop(evd+1=n,A356019_list) ; end if; end if; end do: # R. J. Mathar, Aug 07 2022
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Mathematica
f[n_] := DivisorSum[n, 1 &, EvenQ[DigitCount[#, 2, 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^6] (* Amiram Eldar, Jul 23 2022 *) -
Python
from sympy import divisors from itertools import count, islice def c(n): return bin(n).count("1")&1 == 0 def f(n): return sum(1 for d in divisors(n, generator=True) if c(d)) def agen(): n, adict = 0, dict() for k in count(1): fk = f(k) if fk not in adict: adict[fk] = k while n in adict: yield adict[n]; n += 1 print(list(islice(agen(), 50))) # Michael S. Branicky, Jul 23 2022
Formula
a(n) <= A356040(n). - David A. Corneth, Jul 26 2022
Extensions
More terms from Amiram Eldar, Jul 23 2022
Comments