A302992 -id:A302992 - cp's OEIS frontend

A302991 Decimal expansion of the asymptotic density of abundant numbers.

2, 4, 7, 6, 1, 9, 6

Offset: 0

Amiram Eldar, Apr 17 2018

The existence of this density was asked about by Erich Bessel-Hagen in 1929 and was proved by Harold Davenport in 1933.
Previous evaluated bounds on the value of density, D, were D < 0.47 (Behrend, 1932), 0.241 < D < 0.314 (Behrend, 1933), 0.24432 < D (Salié, 1955), 0.2441 < D < 0.2909 (Wall et al., 1972), and 0.2474 < D < 0.2480 (Deléglise, 1988); the current bounds are 0.2476171 < D < 0.2476475 (Kobayashi, 2010).
0.247619608 < D < 0.247619658 (McNew and Setty, 2025). - Amiram Eldar, Aug 02 2025

			0.2476196...

Felix Behrend, Über numeri abundantes I, S.-Ber. Preuß. Akad. Wiss., math.-nat. Kl. (1932), pp. 322-328.
Erich Bessel-Hagen, Repertorium der höheren Mathematik, 2nd edn., Vol. 1, B. G. Teubner (Leipzig, 1929), pp. 1458-1574.
Harold Davenport, Über numeri abundantes, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, No. 6 (1933), pp. 830-837.

Felix Behrend, Über numeri abundantes, inaugural dissertation, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, pp. 280-293, 1933.
Marc Deléglise, Bounds for the Density of Abundant Integers, Experimental Mathematics, Vol. 7, No. 2 (1998), pp. 137-143.
Mitsuo Kobayashi, On the Density of Abundant Numbers, Ph.D. thesis, Dartmouth College, 2010.
Nathan McNew and Jai Setty, On the densities of covering numbers and abundant numbers, arXiv:2507.23041 [math.NT], 2025.
Hans Salié, Über die Dichte abundanter Zahlen, Mathematische Nachrichten, Vol. 14, No. 1 (1955), pp. 39-46.
Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density Bounds for the Sum of Divisors Function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 773-777; Errata, Vol. 31, No. 138 (1977), p. 616.

Cf. A000203, A005101, A302992.

a(4)-a(5) from Muniru A Asiru, Aug 20 2018

a(6) from Amiram Eldar, Aug 02 2025

A302993 Number of unitary abundant numbers < 10^n.

Original entry on oeis.org

0, 5, 64, 691, 7011, 70030, 699597, 7005450, 70048740, 700321813, 7003128054, 70034216605, 700350142296, 7003426996800, 70033987968599, 700341098675985, 7003416399263312

Offset: 1

Amiram Eldar, Apr 17 2018

Cf. A034448, A034683, A302992.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; uabQ[n_] := usigma[n] > 2 n; c = 0; k = 1; seq={}; Do[While[k < 10^n, If[uabQ[k], c++]; k++]; AppendTo[seq, c], {n, 1, 5}]; seq

Conjecture: Lim_{n->oo} a(n)/10^n = 0.07... is the density of unitary abundant numbers.

a(10)-a(17) from Hiroaki Yamanouchi, Aug 03 2018

A302994 Number of bi-unitary abundant numbers < 10^n.

Original entry on oeis.org

0, 14, 147, 1553, 15450, 155395, 1549818, 15498814, 155079196, 1550331185, 15503061466, 155037242668, 1550370696100, 15503650949671, 155036854371220, 1550366484701654, 15503648102080675

Offset: 1

Amiram Eldar, Apr 17 2018

Cf. A188999, A292982, A302992.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=
DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; babQ[n_] := bsigma[n] > 2 n; c = 0; k = 1; seq={}; Do[While[k < 10^n, If[babQ[k], c++]; k++]; AppendTo[seq, c], {n, 1, 5}]; seq

biusigma(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
a(n) = sum(k=1, 10^n-1, biusigma(k) > 2*k); \\ Michel Marcus, Apr 17 2018

Conjecture: Lim_{n->oo} a(n)/10^n = 0.15... is the density of bi-unitary abundant numbers.

a(8)-a(17) from Hiroaki Yamanouchi, Aug 24 2018

A307821 The number of exponential abundant numbers below 10^n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 30 2019

			Below 10^3 there is only one exponential abundant number, A129575(1) = 900, thus a(3) = 1.

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

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

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]

a(11) from Amiram Eldar, Sep 02 2022

A307823 The number of nonunitary abundant numbers below 10^n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 30 2019

			Below 10^2 there are 5 nonunitary abundant numbers, 36, 48, 72, 80, and 96, thus a(2) = 5.

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

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

Conjecture: Lim_{n->oo} a(n)/10^n = 0.0809... is the density of nonunitary abundant numbers.

A307820 The number of infinitary abundant numbers below 10^n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 30 2019

			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.

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

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

Conjecture: Lim_{n->oo} a(n)/10^n = 0.125... is the density of infinitary abundant numbers.

a(11) from Amiram Eldar, Sep 09 2022

A322287 The number of odd abundant numbers below 10^n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 28 2019

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.

			945 is the only odd abundant number below 10^3, thus a(3) = 1.

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.

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

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

Lim_{n->oo} a(n)/10^n = 0.0020... is the density of odd abundant numbers.

A303736 Number of deficient numbers < 10^n.

Original entry on oeis.org

8, 76, 751, 7508, 75201, 752451, 7523259, 75239327, 752389040, 7523761073, 75238050781, 752382051546, 7523801085404, 75237991990847, 752380354682456, 7523805396029055, 75238037557271886

Offset: 1

Muniru A Asiru, May 19 2018

Limit_{n->oo} a(n)/10^n = 0.752... is the density of deficient numbers (A005100).

Cf. A005100, A302992.

P:=Filtered([1..10^7],n->Sigma(n)<2*n);;
o:=List([1..7],j->[]);; for i in [1..Length(P)] do for n in [1..Length(o)] do if P[i]<10^n then Add(o[n],P[i]); fi; od; od;
a:=List(o,Length);

Accumulate@ Array[Count[Range[10^#, 10^(# + 1) - 1], ?(DivisorSigma[1, #] < 2 # &)] &, 6, 0] (* _Michael De Vlieger, May 20 2018 *)

a(n) = sum(k=1, 10^n-1, sigma(k) < 2*k); \\ Michel Marcus, May 19 2018

a(11)-a(17) from Hiroaki Yamanouchi, Aug 03 2018

A363170 The number of powerful abundant numbers (A363169) not exceeding 10^n.

Original entry on oeis.org

0, 3, 23, 82, 297, 1000, 3268, 10534, 33799, 107901, 343155, 1090189, 3460380, 10970774, 34749182, 109991778, 348006756, 1101058505, 3483105232, 11017518803

Offset: 1

Amiram Eldar, May 19 2023

The ratios a(n)/A118896(n) seem to converge to a positive value as n grows: for n = 14..20 they are 0.506417..., 0.506728..., 0.506863..., 0.506890..., 0.506987..., 0.507059..., 0.507120... .

Conjecture: the asymptotic relative density of the abundant numbers within the powerful numbers exists and equals 0.507... .

			a(2) = 3 since there are 3 powerful abundant numbers not exceeding 10^2: 36, 72 and 100.

Cf. A001694, A005101, A118896, A302992, A363169.

seq[nmax_] := Module[{c = 0, p = 10, k = 1, kmax = 10^nmax, s = {}}, While[k <= kmax, If[DivisorSigma[-1, k] > 2 && Min[FactorInteger[k][[;;, 2]]] > 1, c++]; If[k == p, AppendTo[s, c]; p *= 10]; k++]; s]; seq[5]

is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; } \\ A363169
lista(nmax) = {my(c = 0, p = 10, k = 1, kmax = 10^nmax); while(k <= kmax, if(is(k), c++); if(k == p, print1(c, ", "); p *= 10); k++); }

A306951 Number of weird numbers (A006037) below 10^n.

Original entry on oeis.org

0, 1, 2, 7, 204, 1765, 15554, 173709, 1756426, 17270452, 172559761

Offset: 1

Amiram Eldar, Mar 17 2019

Benkoski and Erdős proved that the asymptotic density of weird numbers is positive, thus lim_{n->oo} a(n)/10^n = c > 0. Is c = 0.0017...?

			The only weird number below 100 is 70 which is larger than 10, thus a(1) = 0 and a(2) = 1.

S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp., 28 (1974), pp. 617-623. Alternate link; 1975 corrigendum.

Cf. A006037, A275487, A275491, A302992.

a(11) from Amiram Eldar, May 26 2023

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A302991 Decimal expansion of the asymptotic density of abundant numbers.

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A302993 Number of unitary abundant numbers < 10^n.

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A302994 Number of bi-unitary abundant numbers < 10^n.

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A307821 The number of exponential abundant numbers below 10^n.

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A307823 The number of nonunitary abundant numbers below 10^n.

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A307820 The number of infinitary abundant numbers below 10^n.

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A322287 The number of odd abundant numbers below 10^n.

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A303736 Number of deficient numbers < 10^n.

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