A360642 a(n) is the least number k such that A093653(k)/A000120(k) = n.

1, 2, 4, 8, 16, 24, 64, 66, 84, 72, 210, 132, 450, 792, 288, 264, 1044, 672, 5328, 528, 1344, 840, 1026, 1056, 4116, 1800, 4128, 2112, 5124, 3780, 6480, 2184, 3360, 8352, 11088, 8448, 4680, 50700, 4200, 4368, 20880, 8280, 13320, 13440, 12420, 4104, 46200, 8736

Offset: 1

Author

Amiram Eldar, Feb 15 2023

Keywords

• nonn
• base

Comments

a(n) exists for all n >= 1 since A093653(2^(k-1))/A000120(2^(k-1)) = k for all k >= 1.

Analogous to A007539 as A175522 is analogous to perfect numbers (A000396).

Examples

			a(1) = 1 since A093653(1)/A000120(1) = 1/1 = 1.
a(2) = 2 since A093653(2)/A000120(2) = 2/1 = 2, and 2 is the least number with this property.
a(3) = 4 since A093653(4)/A000120(4) = 3/1 = 3, and 4 is the least number with this property.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000120, A000396, A007539, A093653, A093687, A175522, A175526, A360641.

Programs

Mathematica

seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n <= nmax, i = DivisorSum[n, DigitCount[#, 2, 1] &]/DigitCount[n, 2, 1]; If[IntegerQ[i] && i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[50, 10^5]

PARI

lista(len, nmax) = {my(s = vector(len), c = 0, n = 1, i); while(c < len && n <= nmax, i = sumdiv(n, d, hammingweight(d))/hammingweight(n); if(denominator(i) == 1 && i <= len && s[i] == 0, c++; s[i] = n); n++); s }

Python

# uses imports and definitions in A093653, A000120
from itertools import count, islice
def f(n): q, r = divmod(A093653(n), A000120(n)); return q if r == 0 else 0
def agen():
    n, adict = 1, dict()
    for k in count(1):
        v = f(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 48))) # Michael S. Branicky, Feb 15 2023

Formula

a(n) <= 2^(n-1).