A152573 The smallest superharmonic number with index n.

6, 30, 102, 186, 1146, 762, 8022, 5334, 35526, 23622, 145542, 49146, 540606, 344022, 2309862, 786426, 4718586, 3145722, 33030102, 22020054, 146276166, 97517382, 599260422, 399506694, 4194822954, 2796546858, 18577073082, 6441615366, 38649937926, 12884901882

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..30.

Graeme L. Cohen, Superharmonic numbers, Math. Comp., Vol. 78, No. 265 (2009), pp. 421-429.

Formula

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

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A000005, A000203, A001599.

Sequence in context: A008525 A101375 A074007 * A348257 A353894 A353883

Adjacent sequences: A152570 A152571 A152572 * A152574 A152575 A152576

Keyword

nonn

Author

R. J. Mathar, Dec 08 2008

Extensions

a(25)-a(30) from Amiram Eldar, Jun 03 2020

Status

approved