A383758
Least integer k for which sigma(k - x) + sigma(k + x) = n*k has at least one solution.
Original entry on oeis.org
1, 2, 6, 24, 93, 1952, 14412, 361881, 61824672
Offset: 2
a(4) = 6 because the equation sigma(6-x) + sigma(6+x) = 4*6 has the solution x = 0 and no smaller number possesses this property. See A000396, A383268, and A383269.
a(5) = 24 because the equation sigma(24-x) + sigma(24+x) = 5*24 has the solution x = 0. This is verified as follows: sigma(24-0) + sigma(24+0) = sigma(24) + sigma(24) = 60 + 60 = 120 = 5*24. Moreover, no smaller number possesses this property. See A141643.
a(6) = 93 because the equation sigma(93 - x) + sigma(93 + x) = 6 * 93 has the solution x = 87: sigma(93 - 87) + sigma(93 + 87) = sigma(6) + sigma(180) = 12 + 546 = 6*93. Moreover, no smaller number possesses this property.
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isok(k,n) = forstep(x=k-1, 0, -1, if (sigma(k - x) + sigma(k + x) == n*k, return(1)));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, May 10 2025
A383920
Smallest m such that sigma(m) >= n*m/2.
Original entry on oeis.org
1, 2, 6, 24, 120, 1680, 27720, 720720, 122522400, 41902660800, 130429015516800, 3066842656354276800, 1970992304700453905270400, 168721307030313765796546413936000, 1897544233056092162003806758651798777216000, 8201519488959040182625924708238885435575055666675808000
Offset: 2
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
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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)
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(* 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 *)
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a(n) = my(k=1); while (sigma(k) < k*n/2, k++); k;
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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])););
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