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 A354424 Numbers k for which the ratio A008475(k)/k reaches a record low. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 10 04:38 EST 2023. Contains 367699 sequences. (Running on oeis4.)