A173490 -id:A173490 - cp's OEIS frontend

A007621 Impractical numbers: even abundant numbers (A173490) that are not practical(2) (A007620).

70, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 350, 354, 366, 372, 402, 426, 438, 444, 474, 490, 492, 498, 516, 534, 550, 564, 572, 582, 606, 618, 636, 642, 650, 654, 678, 708, 732, 738, 748, 762, 770, 774, 786, 804, 822, 834, 836, 846, 852, 876, 894

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Robert G. Wilson v, Letter to N. J. A. Sloane, date unknown.

A173490 with A007620 removed. Cf. A035480.

practicalQ[n_] := Module[{d = Divisors[n], c}, c = Rest[CoefficientList[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], x]]; Flatten[Position[c, 0]] == {} && c[[-1]] > 1]; Select[2*Range[500], DivisorSigma[1, #] > 2 # && ! practicalQ[#] &] (* T. D. Noe, Jun 25 2012 *)

More terms from David W. Wilson.

A005101 Abundant numbers (sum of divisors of m exceeds 2m).

Original entry on oeis.org

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270

Offset: 1

N. J. A. Sloane

A number m is abundant if sigma(m) > 2m (this sequence), perfect if sigma(m) = 2m (cf. A000396), or deficient if sigma(m) < 2m (cf. A005100), where sigma(m) is the sum of the divisors of m (A000203).
While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!
It appears that for m abundant and > 23, 2*A001055(m) - A101113(m) is NOT 0. - Eric Desbiaux, Jun 01 2009
If m is a term so is every positive multiple of m. "Primitive" terms are in A091191.
If m=6k (k>=2), then sigma(m) >= 1 + k + 2*k + 3*k + 6*k > 12*k = 2*m. Thus all such m are in the sequence.
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Thus the n-th abundant number is asymptotic to 4.0322*n < n/A(2) < 4.0421*n. - Daniel Forgues, Oct 11 2015
From Bob Selcoe, Mar 28 2017 (prompted by correspondence with Peter Seymour): (Start)
Applying similar logic as the proof that all multiples of 6 >= 12 appear in the sequence, for all odd primes p:
i) all numbers of the form j*p*2^k (j >= 1) appear in the sequence when p < 2^(k+1) - 1;
ii) no numbers appear when p > 2^(k+1) - 1 (i.e., are deficient and are in A005100);
iii) when p = 2^(k+1) - 1 (i.e., perfect numbers, A000396), j*p*2^k (j >= 2) appear.
Note that redundancies are eliminated when evaluating p only in the interval [2^k, 2^(k+1)].
The first few even terms not of the forms i or iii are {70, 350, 490, 550, 572, 650, 770, ...}. (End)

L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., Vol. 44 (1913), pp. 264-296.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Britton, Perfect Number Analyser.
C. K. Caldwell, The Prime Glossary, abundant number.
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math., Volume 7, Issue 2 (1998), pp. 137-143.
Jason Earls, On Smarandache repunit n numbers, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), page 243.
S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 112.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Walter Nissen, Abundancy : Some Resources .
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
Eric Weisstein's World of Mathematics, Abundant Number.
Eric Weisstein's World of Mathematics, Abundance.
Wikipedia, Abundant number.
Index entries for "core" sequences.

Cf. A005835, A005100, A091194, A091196, A080224, A091191 (primitive).
Cf. A005231 and A006038 (odd abundant numbers).
Cf. A094268 (n consecutive abundant numbers).
Cf. A173490 (even abundant numbers).
Cf. A001065.
Cf. A000396 (perfect numbers).
Cf. A302991.

a005101 n = a005101_list !! (n-1)
a005101_list = filter (\x -> a001065 x > x) [1..]
-- Reinhard Zumkeller, Nov 01 2015, Jan 21 2013

with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d,`,n) fi: od:
isA005101 := proc(n)
    simplify(numtheory[sigma](n) > 2*n) ;
end proc: # R. J. Mathar, Jun 18 2015
A005101 := proc(n)
    option remember ;
    local a ;
    if n =1 then
        12 ;
    else
        a := procname(n-1)+1 ;
        while numtheory[sigma](a) <= 2*a do
            a := a+1 ;
        end do ;
        a ;
    end if ;
end proc: # R. J. Mathar, Oct 11 2017

abQ[n_] := DivisorSigma[1, n] > 2n; A005101 = Select[ Range[270], abQ[ # ] &] (* Robert G. Wilson v, Sep 15 2005 *)
Select[Range[300], DivisorSigma[1, #] > 2 # &] (* Vincenzo Librandi, Oct 12 2015 *)

isA005101(n) = (sigma(n) > 2*n) \\ Michael B. Porter, Nov 07 2009

from sympy import divisors
def ok(n): return sum(divisors(n)) > 2*n
print(list(filter(ok, range(1, 271)))) # Michael S. Branicky, Aug 29 2021

from sympy import divisor_sigma
from itertools import count, islice
def A005101_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) > 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue
A005101_list = list(islice(A005101_gen(), 20)) # Chai Wah Wu, Jan 14 2022

a(n) is asymptotic to C*n with C=4.038... (Deléglise, 1998). - Benoit Cloitre, Sep 04 2002
A005101 = { n | A033880(n) > 0 }. - M. F. Hasler, Apr 19 2012
A001065(a(n)) > a(n). - Reinhard Zumkeller, Nov 01 2015

A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

Original entry on oeis.org

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955

Offset: 1

N. J. A. Sloane

nonn

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number.
Schiffman notes that 945+630k is in this sequence for all k < 52. Most of the first initial terms are of the form. Among the 1996 terms below 10^6, 1164 terms are of that form, and only 26 terms are not divisible by 5, cf. A064001. - M. F. Hasler, Jul 16 2016
From M. F. Hasler, Jul 28 2016: (Start)
Any multiple of an abundant number is again abundant, see A006038 for primitive terms, i.e., those which are not a multiple of an earlier term.
An odd abundant number must have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. To see this, write the relative abundancy A(N) = sigma(N)/N = sigma[-1](N) as A(Product p_i^e_i) = Product (p_i-1/p_i^e_i)/(p_i-1) < Product p_i/(p_i-1).
See A115414 for terms not divisible by 3, A064001 for terms not divisible by 5, A112640 for terms coprime to 5*7, and A047802 for other generalizations.
As of today, we don't know of a difference between this set S of odd abundant numbers and the set S' of odd semiperfect numbers: Elements of S' \ S would be perfect (A000396), and elements of S \ S' would be weird (A006037), but no odd weird or perfect number is known. (End)
For any term m in this sequence, A064989(m) is also an abundant number (in A005101), and for any term x in A115414, A064989(x) is in this sequence. If there are no odd perfect numbers, then applying A064989 to these terms and sorting into ascending order gives A337386. - Antti Karttunen, Aug 28 2020
There exist infinitely many terms m such that 2*m+1 is also a term. An example of such a term is given by m = 985571808130707987847768908867571007187. - Max Alekseyev, Nov 16 2023

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 13.
R. K. Guy, Unsolved Problems in Number Theory, B2.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

Metin Sariyar, Table of n, a(n) for n = 1..32000 (terms 1..1000 from T. D. Noe)
Jill Britton, Perfect Number Analyzer.
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
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.
Victor Meally, Letter to N. J. A. Sloane, no date.
Walter Nissen, Abundancy : Some Resources.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.
Jay L. Schiffman and Christopher S. Simons, More Odd Abundant Sequences, Volume 38, Number 1 (September 2005), pp. 7-8.

Cf. A000203, A005835, A006038, A115414, A064001, A112640, A122036, A136446, A005101, A173490, A039725, A064989, A337386.

A005231 := proc(n) option remember ; local a ; if n = 1 then 945 ; else for a from procname(n-1)+2 by 2 do if numtheory[sigma](a) > 2*a then return a; end if; end do: end if; end proc: # R. J. Mathar, Mar 20 2011

fQ[n_] := DivisorSigma[1, n] > 2n; Select[1 + 2Range@ 9000, fQ] (* Robert G. Wilson v, Mar 20 2011 *)

je=[]; forstep(n=1,15000,2, if(sigma(n)>2*n, je=concat(je,n))); je

is_A005231(n)={bittest(n,0)&&sigma(n)>2*n} \\ M. F. Hasler, Jul 28 2016

list(lim)=my(v=List()); forfactored(n=945,lim\1, if(n[2][1,1]>2 && sigma(n,-1)>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Apr 21 2022

a(n) ~ k*n for some constant k (perhaps around 500). - Charles R Greathouse IV, Apr 21 2022

482.8 < k < 489.8 (based on density bounds by Kobayashi et al., 2009). - Amiram Eldar, Jul 17 2022

More terms from James Sellers

A353061 Zumkeller numbers (A083207) that are not practical numbers (A005153).

Original entry on oeis.org

70, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 350, 354, 366, 372, 402, 426, 438, 444, 474, 490, 492, 498, 516, 534, 550, 564, 572, 582, 606, 618, 636, 642, 650, 654, 678, 708, 732, 762, 770, 786, 804, 822, 834, 836, 852, 876, 894, 906, 910, 940, 942, 945, 948, 978, 996

Offset: 1

Jianing Song, Apr 20 2022

nonn

Different from A007621: A007621 contains no odd numbers, while every odd term in A083207 is here. The numbers 738, 748, 774, 846, ... are in A007621 and are not here.
But the subsequence of even terms (A005843 intersect this sequence) is a subsequence of A007621:
- A007621 = A173490 \ A007620;
- A005843 intersect this sequence = (A005843 intersect A083207) \ A005153;
- A083207 is a subsequence of A023196, and every perfect number is practical;
- So, (A005843 intersect A083207) \ A005153 is a subsequence of A173490, and A005153 is a supersequence of A007620.

			70 is a term since 70 is a Zumkeller number but not a practical number: 1+5+7+10+14+35 = 2+70, so 70 is a Zumkeller number; but 4 cannot be written as a sum of distinct divisors of 70, so 70 is not practical.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A083207, A005153, A007621.

Cf. also A173490, A007620, A023196, A005843.

isA353061(n) = is(n) && !is_A005153(n) \\ See A083207 for is(n) and A005153 for is_A005153(n)

A039725 Even abundant numbers divided by 2.

Original entry on oeis.org

6, 9, 10, 12, 15, 18, 20, 21, 24, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 60, 63, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 88, 90, 93, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 117, 120, 123, 126, 129, 130, 132, 135, 136

Offset: 1

N. J. A. Sloane. This was included in the 1973 "Handbook", but was then dropped from the database. Resubmitted by James Sellers. Entry revised by N. J. A. Sloane, Jun 12 2012

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!

The asymptotic density of this sequence is in the interval (0.491096, 0.491156) (based on the known bounds on the densities of A005101 and A005231; see A302991 and A322287). - Amiram Eldar, Mar 11 2024

			The first even abundant number is 12, so 12/2 = 6 is the first element in this sequence.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A005101, A005231, A173490, A302991, A322287.

Select[Range[2,300,2],DivisorSigma[1,#]>2*#&]/2 (* Harvey P. Dale, Mar 22 2020 *)

a(n) = A173490(n) / 2.

Corrected and edited by Daniel Forgues, Nov 22 2010

A383398 a(n) is the smallest number whose sum with any previous term is abundant.

Original entry on oeis.org

1, 11, 19, 29, 59, 349, 521, 2071, 66949, 223231, 3660191, 4552181, 5500081, 10161979, 12235619, 47859629

Offset: 1

Jakub Buczak, Apr 25 2025

The terms are generally either prime or semiprime. This results in all known terms to be deficient (see A005100).
If a(1) is an even abundant number, then the set of all the terms is simply the set of all the even abundant numbers (see A173490).
I conjecture that all the terms are odd integers ending in 1 or 9. The odd nature of the terms seems particularly likely, as the sum of a(n) that's even with any previous term would need to be an odd abundant number (see A005231).
This is also equivalent to the sum of any 2 terms being an abundant number.

			29 is a member, because 29+19, 29+11 and 29+1 are all abundant numbers.

Cf. A383399, A005101, A005231, A005100, A173490.

Cf. A000040, A001358.

q:= n-> is(numtheory[sigma](n)>2*n):
a:= proc(n) option remember; local k, l;
      l:= [seq(a(i), i=1..n-1)]:
      for k while not andmap(j-> q(k+j), l) do od; k
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Apr 25 2025

a[1] = 1; a[n_] := a[n] = Module[{k = a[n-1] + 1}, While[AnyTrue[Array[a, n-1], DivisorSigma[-1, #+k] <= 2 &], k++]; k]; Array[a, 10] (* Amiram Eldar, Apr 26 2025 *)

isabundant(n) = (sigma(n) > 2*n) ;
isok(k, n, va) = for (i=1, n-1, if (! isabundant(k+va[i]), return(0));); return(1);
lista(nn) = my(va=vector(nn)); for (n=1, nn, my(k=if (n==1, 1, 1+va[n-1])); while(! isok(k, n, va), k++); k; va[n] = k;); va; \\ Michel Marcus, Apr 26 2025

a(16) from Amiram Eldar, Apr 26 2025

A383650 Averages k of a twin prime pair such that 3k2^d is also the average of a twin prime pair for some divisor d of 3*k.

Original entry on oeis.org

4, 6, 12, 18, 30, 60, 72, 108, 138, 192, 240, 270, 312, 348, 420, 432, 570, 642, 810, 822, 828, 1020, 1050, 1092, 1302, 1320, 1452, 1620, 1668, 1698, 1722, 1950, 1998, 2310, 2550, 2688, 2712, 2730, 2970, 3000, 3168, 3258, 3330, 3372, 3462, 3468, 3540, 3582, 4092

Offset: 1

Juri-Stepan Gerasimov, May 04 2025

nonn

All terms after a(2) are abundant (A005101), this is because all primes greater than 3 are of the form 6*k +- 1, thus the average of twin primes is 6*k, and since any multiple of a perfect or abundant number is abundant itself, it means that this property holds for all n > 11. - Jakub Buczak, May 04 2025

			Average 4 of a twin prime pair is in the sequence because 3*4*2^4 = 192 is also the average of twin primes 191 and 193 for divisor d = 4 of 3*k = 3*4 = 12.

Jakub Buczak, Table of n, a(n) for n = 1..125

Supersequence of A014574.

Cf. A000040, A383475, A005101, A173490.

[k: k in [4..4100] | not #[d: d in Divisors(3*k) | IsPrime(k-1) and IsPrime(k+1) and IsPrime(3*k*2^d-1) and IsPrime(3*k*2^d+1)] eq 0];

a(n) ~ b*n^c for some constants b and c as n tends to infinity (conjectured). - Jakub Buczak, May 04 2025

More terms from Jakub Buczak, May 04 2025

cp's OEIS Frontend

A007621 Impractical numbers: even abundant numbers (A173490) that are not practical(2) (A007620).

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A005101 Abundant numbers (sum of divisors of m exceeds 2m).

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A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

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A353061 Zumkeller numbers (A083207) that are not practical numbers (A005153).

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A039725 Even abundant numbers divided by 2.

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A383398 a(n) is the smallest number whose sum with any previous term is abundant.

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A383650 Averages k of a twin prime pair such that 3*k*2^d is also the average of a twin prime pair for some divisor d of 3*k.

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A383650 Averages k of a twin prime pair such that 3k2^d is also the average of a twin prime pair for some divisor d of 3*k.