A302993 -id:A302993

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A034683

Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

+10
39

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 210, 222, 246, 258, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822, 834

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If a term n in the sequence ends in neither 0 nor 5, then 10*n is also in the sequence. - Lekraj Beedassy, Jun 11 2004

The lower asymptotic density of this sequence is larger than 1/18 = 0.0555... which is the density of its subsequence of numbers of the form 6*m where gcd(m, 6) = 1 and m > 1. Numerically, based on counts of terms below 10^n (A302993), it seems that this sequence has an asymptotic density which equals to about 0.070034... - Amiram Eldar, Feb 13 2021

The asymptotic density of this sequence is in the interval (0.0674, 0.1055) (Wall, 1970). - Amiram Eldar, Apr 18 2024

All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, usigma(k)/k < 15/Pi^2 = 1.519817... (A082020; the record values are attained at the squares of primorials, A061742). - Amiram Eldar, Jul 20 2024

REFERENCES

C. Sung, Mathematical Buds, "Unitary Divisors", Chap. V, pp. 42-67, Ed. H. D. Ruderman, Mu Alpha Theta OK 1978.

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

MAPLE

isA034683 := proc(n)

is(A034448(n) > 2*n) ;

end proc:

for n from 1 do

if isA034683(n) then

print(n);

end if;

end do: # R. J. Mathar, Nov 10 2014

MATHEMATICA

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];

Select[Range[1000], usigma[#] > 2#&] (* Jean-François Alcover, Mar 23 2020, after Giovanni Resta in A034448 *)

PROG

(PARI) is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]) > 2*n; } \\ Amiram Eldar, Apr 18 2024

CROSSREFS

Subsequence of A005101.

Cf. A034444, A034448, A129487, A302993.

Cf. A001694, A061742, A082020.

KEYWORD

nonn

AUTHOR

Erich Friedman

STATUS

approved

A307821

The number of exponential abundant numbers below 10^n.

+10
5

0, 0, 1, 12, 102, 1045, 10449, 104365, 1043641, 10436775, 104367354

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..11.

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

CROSSREFS

Cf. A051377, A129575, A302992, A302993, A302994, A307820, A307822, A307823.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Apr 30 2019

EXTENSIONS

a(11) from Amiram Eldar, Sep 02 2022

STATUS

approved

A307823

The number of nonunitary abundant numbers below 10^n.

+10
5

0, 5, 75, 812, 8079, 81052, 808477, 8097357, 80939927, 809350234

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..10.

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

CROSSREFS

Cf. A048146, A064597, A302992, A302993, A302994, A307820, A307821.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Apr 30 2019

STATUS

approved

A307820

The number of infinitary abundant numbers below 10^n.

+10
4

0, 12, 114, 1270, 12518, 125634, 1257749, 12570993, 125716733, 1256921422, 12570417639

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..11.

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

CROSSREFS

Cf. A049417, A129656, A302992, A302993, A302994, A307821, A307823.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Apr 30 2019

EXTENSIONS

a(11) from Amiram Eldar, Sep 09 2022

STATUS

approved

A322287

The number of odd abundant numbers below 10^n.

+10
3

0, 0, 1, 23, 210, 1996, 20661, 205366, 2048662, 20502004, 204951472

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

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.

LINKS

Table of n, a(n) for n=1..11.

C. W. Anderson, Density of Deficient Odd Numbers, The American Mathematical Monthly, Vol. 82, No. 10 (1975), pp. 1018-1020.

Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, On the distribution of sociable numbers, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.

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

CROSSREFS

Cf. A000203, A005231, A302992, A302993, A302994, A307820, A307821, A307823.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Aug 28 2019

STATUS

approved

A308054

The number of coreful abundant numbers (A308053) below 10^n.

+10
0

0, 1, 24, 259, 2614, 26222, 262220, 2622178, 26221610, 262215860, 2622158194

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..11.

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

CROSSREFS

Cf. A302992, A302993, A302994, A307820, A307821, A307823, A307961, A308053.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, May 10 2019

EXTENSIONS

a(11) from Amiram Eldar, Sep 02 2022

STATUS

approved

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