A152573 - OEIS

%I #12 Jun 03 2020 10:06:14

%S 6,30,102,186,1146,762,8022,5334,35526,23622,145542,49146,540606,

%T 344022,2309862,786426,4718586,3145722,33030102,22020054,146276166,

%U 97517382,599260422,399506694,4194822954,2796546858,18577073082,6441615366,38649937926,12884901882

%N The smallest superharmonic number with index n.

%C a(n) is the smallest number s such that sigma(s) divides s^n*tau(s). In short: a(n) is the smallest superharmonic number with index n.

%H Graeme L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-08-02147-9">Superharmonic numbers</a>, Math. Comp., Vol. 78, No. 265 (2009), pp. 421-429.

%F a(n) = min s: A000203(s) | s^n*A000005(s).

%e For n = 1, a(1) = 6 because sigma(6) = 12 divides 6^1*tau(6) = 6*4 = 24.

%t ind[n_] := Module[{d = Denominator[DivisorSigma[0, n]/DivisorSigma[1, n]], m, p, e, en},  m = 0; Do[{p, e} = pe; en = IntegerExponent[n, p]; If[en == 0, m = 0; Break[], m = Max[m, Ceiling[e/en]]], {pe, FactorInteger[d]}]; m]; mx = 14; c = 0; n = 1; v = Table[0, {mx}]; While[c < mx, n++; i = ind[n]; If[i > 0 && i <= mx && v[[i]] == 0, c++; v[[i]] = n]]; v (* _Amiram Eldar_, Jun 03 2020 *)

%o (PARI) isharmonic(val, index) = ((val^index*numdiv(val) % sigma(val)) == 0)

%o a(n) = {val = 2; ok = 0; until (ok, if (isharmonic(val, n), if (n == 1, ok = 1, indi = 1; while (! isharmonic(val, indi), indi++); if (indi == n, ok = 1););); if (! ok, val++);); return (val);} \\ _Michel Marcus_, Jul 24 2013

%Y Cf. A000005, A000203, A001599.

%K nonn

%O 1,1

%A _R. J. Mathar_, Dec 08 2008

%E a(25)-a(30) from _Amiram Eldar_, Jun 03 2020