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

1, 2, 6, 24, 120, 1680, 27720, 720720, 122522400, 41902660800, 130429015516800, 3066842656354276800, 1970992304700453905270400, 168721307030313765796546413936000, 1897544233056092162003806758651798777216000, 8201519488959040182625924708238885435575055666675808000

Offset: 2

Author

Michel Marcus, May 15 2025

Keywords

• nonn

Comments

For n>= 11, terms were computed with 2nd PARI program using the T. D. Noe algorithm.

Examples

			From _Michael De Vlieger_, May 22 2025: (Start)
Table of a(n), n = 2..10, showing prime power decomposition:
                                         Prime power
                                         factor exponent
                                             111
 n     m = a(n)    sigma(m)      n*m/2   2357137
------------------------------------------------
 2           1           1           1   0
 3           2           3           3   1
 4           6          12          12   11
 5          24          60          60   31
 6         120         360         360   311
 7        1680        5952        5880   4111
 8       27720      112320      110880   32111
 9      720720     3249792     3243240   421111
10   122522400   614210688   612612000   5221111 (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..27
• T. D. Noe, An algorithm for finding the least k with sigma(k) >= nk

Crossrefs

Cf. A000203, A023199, A317681.

Cf. A004490. Subsequence of A004394.

Programs

Mathematica

(* First, load function f from A025487, then *)
nn = 12; s = Union@ Flatten@ f[nn + 4]; m = Length[s];
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 *)

PARI

a(n) = my(k=1); while (sigma(k) < k*n/2, k++); k;

PARI

ab(x) = sigma(x)/x;
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)););
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])););