cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A034683 Unitary abundant numbers: numbers k such that usigma(k) > 2*k.

Original entry on oeis.org

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
Offset: 1

Views

Author

Erich Friedman

Keywords

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

Crossrefs

Subsequence of A005101.
Cf. A034444, A034448, A129487, A302993.
Cf. A001694, A061742, A082020.

Programs

  • 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 *)
  • 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

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

Views

Author

Amiram Eldar, Apr 30 2019

Keywords

Examples

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

Crossrefs

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

Programs

  • 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

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]

Extensions

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

Views

Author

Amiram Eldar, Apr 30 2019

Keywords

Examples

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

Crossrefs

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

Programs

  • 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

Formula

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

Views

Author

Amiram Eldar, Apr 30 2019

Keywords

Examples

			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.

Crossrefs

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

Programs

  • 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

Formula

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

Extensions

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

Views

Author

Amiram Eldar, Aug 28 2019

Keywords

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.

Examples

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

Links

Crossrefs

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

Programs

  • 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

Formula

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

A308054 The number of coreful abundant numbers (A308053) below 10^n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, May 10 2019

Keywords

Examples

			Below 10^2 there is only one coreful abundant number, 72, hence a(2) = 1.

Crossrefs

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

Programs

  • 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

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]

Extensions

a(11) from Amiram Eldar, Sep 02 2022
Showing 1-6 of 6 results.

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