A354424 Numbers k for which the ratio A008475(k)/k reaches a record low.

2, 6, 10, 12, 15, 20, 28, 30, 40, 42, 56, 60, 84, 105, 120, 140, 168, 180, 210, 252, 280, 315, 330, 360, 385, 390, 420, 616, 630, 660, 770, 780, 840, 924, 1092, 1155, 1260, 1540, 1820, 1848, 1980, 2184, 2310, 2520, 2730, 3080, 3465, 3640, 3960, 4095, 4290, 4620, 5460, 6552, 6930

Offset

1,1

Comments

Sequence gives the numbers k for which m/k reaches a record low, where m is minimal so that the symmetric group S_m has an element of order k.

Links

Table of n, a(n) for n=1..55.

Example

First, an element of order 2 shows up in S_2, so the smallest ratio we've seen so far is 1. This is the smallest ratio we see until we reach 6, since there's an element of order 6 in S_5. Next is 10, since there's an element of order 10 in S_7, and 7/10 is the next ratio smaller than 5/6. Then comes 12, since S_7 also has an element of order 12, and 7/12 is the next ratio less than 7/10, etc.

Mathematica

s = {}; fm = 2; Do[If[(f = Plus @@ Power @@@ FactorInteger[n]/n) < fm, fm = f; AppendTo[s, n]], {n, 2, 7000}]; s (* Amiram Eldar, Jul 12 2022 *)

Prog

(Sage)
memo = {1: (2, 1)}
def a(n):
    if n in memo.keys(): return memo[n]
    _ = a(n-1)
    prev, prevRatio = memo[n-1]
    ratio = 1
    N = prev
    while ratio >= prevRatio:
        N += 1
        # compute m so that S_m has an element of order N
        principalDivisors = list(factor(N))
        m = sum([a^b for (a, b) in principalDivisors])
        ratio = m/N
    memo[n] = (N, ratio)
    return N
(PARI) b(n) = my(f=factor(n)); vecsum(vector(#f~, i, f[i, 1]^f[i, 2])); \\ A008475
lista(nn) = my(m=oo, list=List(), x); for (n=2, nn, if ((x=b(n)/n) < m, m = x; listput(list, n); ); ); Vec(list); \\ Michel Marcus, Jul 12 2022

Crossrefs

Cf. A008475.

Sequence in context: A195064 A328926 A055743 * A189680 A189395 A190003

Adjacent sequences: A354421 A354422 A354423 * A354425 A354426 A354427

Keyword

nonn

Author

Chris Grossack, Jul 11 2022

Status

approved