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A383920 Smallest m such that sigma(m) >= n*m/2.

%I A383920 #22 May 22 2025 11:45:55
%S A383920 1,2,6,24,120,1680,27720,720720,122522400,41902660800,130429015516800,
%T A383920 3066842656354276800,1970992304700453905270400,
%U A383920 168721307030313765796546413936000,1897544233056092162003806758651798777216000,8201519488959040182625924708238885435575055666675808000
%N A383920 Smallest m such that sigma(m) >= n*m/2.
%C A383920 For n>= 11, terms were computed with 2nd PARI program using the T. D. Noe algorithm.
%H A383920 Michael De Vlieger, <a href="/A383920/b383920.txt">Table of n, a(n) for n = 2..27</a>
%H A383920 T. D. Noe, <a href="http://www.sspectra.com/math/A023199.pdf">An algorithm for finding the least k with sigma(k) >= nk</a>
%e A383920 From _Michael De Vlieger_, May 22 2025: (Start)
%e A383920 Table of a(n), n = 2..10, showing prime power decomposition:
%e A383920                                          Prime power
%e A383920                                          factor exponent
%e A383920                                              111
%e A383920  n     m = a(n)    sigma(m)      n*m/2   2357137
%e A383920 ------------------------------------------------
%e A383920  2           1           1           1   0
%e A383920  3           2           3           3   1
%e A383920  4           6          12          12   11
%e A383920  5          24          60          60   31
%e A383920  6         120         360         360   311
%e A383920  7        1680        5952        5880   4111
%e A383920  8       27720      112320      110880   32111
%e A383920  9      720720     3249792     3243240   421111
%e A383920 10   122522400   614210688   612612000   5221111 (End)
%t A383920 (* First, load function f from A025487, then *)
%t A383920 nn = 12; s = Union@ Flatten@ f[nn + 4]; m = Length[s];
%t A383920 Monitor[Reap[Do[k = 1; While[And[DivisorSigma[1, #] < n*#/2 &[ s[[k]] ], k < m], k++]; If[k == m, Break[], Sow[s[[k]] ] ], {n, 2, nn}] ][[-1, 1]], n] (* _Michael De Vlieger_, May 21 2025 *)
%o A383920 (PARI) a(n) = my(k=1); while (sigma(k) < k*n/2, k++); k;
%o A383920 (PARI) ab(x) = sigma(x)/x;
%o A383920 findpos(vca, val) = for (i=1, #vca -1, if ((sigma(vca[i])/vca[i] < val) && (sigma(vca[i+1])/vca[i+1] > val), return(i)););
%o A383920 a(n) = if (n==1, return(0)); if (n==2, return(1)); my(val = n/2, vca = readvec("c:/gp/bfiles/b004490.txt"), vsa = readvec("c:/gp/bfiles/b004394.txt"), wc = select(x->(ab(x) == val), vca)); if (#wc, return(wc[1])); my(ipos = findpos(vca, val), c1 = vca[ipos], c2 = vca[ipos+1], ws = select(x->((x>c1) && (x<=c2)), vsa)); for (i=1, #ws, if (ab(ws[i]) >= val, return(ws[i])););
%Y A383920 Cf. A000203, A023199, A317681.
%Y A383920 Cf. A004490. Subsequence of A004394.
%K A383920 nonn
%O A383920 2,2
%A A383920 _Michel Marcus_, May 15 2025