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The number of exponential abundant numbers below 10^n.
+10
5
0, 0, 1, 12, 102, 1045, 10449, 104365, 1043641, 10436775, 104367354
FORMULA
Limit_{n->oo} a(n)/10^n = 0.001043673... is the density of exponential abundant numbers (see A129575). [Updated by Amiram Eldar, Sep 02 2022]
EXAMPLE
Below 10^3 there is only one exponential abundant number, A129575(1) = 900, thus a(3) = 1.
MATHEMATICA
fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ esigma[k]>2k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq
The number of nonunitary abundant numbers below 10^n.
+10
5
0, 5, 75, 812, 8079, 81052, 808477, 8097357, 80939927, 809350234
FORMULA
Conjecture: Lim_{n->oo} a(n)/10^n = 0.0809... is the density of nonunitary abundant numbers.
EXAMPLE
Below 10^2 there are 5 nonunitary abundant numbers, 36, 48, 72, 80, and 96, thus a(2) = 5.
MATHEMATICA
usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ nusigma[k]>k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq
The number of infinitary abundant numbers below 10^n.
+10
4
0, 12, 114, 1270, 12518, 125634, 1257749, 12570993, 125716733, 1256921422, 12570417639
FORMULA
Conjecture: Lim_{n->oo} a(n)/10^n = 0.125... is the density of infinitary abundant numbers.
EXAMPLE
Below 10^2 there are 12 infinitary abundant numbers, 24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, and 96, thus a(2) = 12.
MATHEMATICA
fun[p_, e_] := Module[{ b = IntegerDigits[e, 2]}, m=Length[b]; Product[If[b[[j]] > 0, 1+p^(2^(m-j)), 1], {j, 1, m}]]; isigma[1]=1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; c = 0; k = 1; seq={}; Do[ While[ k < 10^n, If[ isigma[k]>2k, c++ ]; k ++]; AppendTo[seq, c], {n, 1, 5}]; seq
The number of odd abundant numbers below 10^n.
+10
3
0, 0, 1, 23, 210, 1996, 20661, 205366, 2048662, 20502004, 204951472
COMMENTS
Anderson proved that the density of odd deficient numbers is at least (48 - 3*Pi^2)/(32 - Pi^2) ~ 0.831...
Kobayashi et al. proved that the density of odd abundant numbers is between 0.002042 and 0.002071.
FORMULA
Lim_{n->oo} a(n)/10^n = 0.0020... is the density of odd abundant numbers.
EXAMPLE
945 is the only odd abundant number below 10^3, thus a(3) = 1.
MATHEMATICA
abQ[n_] := DivisorSigma[1, n] > 2 n; c = 0; k = 1; s = {}; Do[While[k < 10^n, If[abQ[k], c++]; k += 2]; AppendTo[s, c], {n, 1, 5}]; s
The number of coreful abundant numbers ( A308053) below 10^n.
+10
0
0, 1, 24, 259, 2614, 26222, 262220, 2622178, 26221610, 262215860, 2622158194
FORMULA
a(n) ~ c * 10^n, where c = 0.0262215... is the asymptotic density of the coreful abundant numbers (see A308053). [Updated by Amiram Eldar, Sep 02 2022]
EXAMPLE
Below 10^2 there is only one coreful abundant number, 72, hence a(2) = 1.
MATHEMATICA
f[p_, e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); cpQ[n_] := csigma[n] > 2*n; s={0}; c=0; p=100; Do[If[k==p, AppendTo[s, c]; p*=10]; If[cpQ[k], c++], {k, 1, 1000001}]; s
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