A002093 -id:A002093 - cp's OEIS frontend

A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144

Offset: 1

N. J. A. Sloane

Multiplicative: If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (this sequence) (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100).
a(n) is the number of sublattices of index n in a generic 2-dimensional lattice. - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001 [In the language of group theory, a(n) is the number of index-n subgroups of Z x Z. - Jianing Song, Nov 05 2022]
The sublattices of index n are in one-to-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} d = sigma(n), which is a(n). A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * Product_{p|n} (1+1/p), which is A001615. [Cf. Grady reference.]
Sum of number of common divisors of n and m, where m runs from 1 to n. - Naohiro Nomoto, Jan 10 2004
a(n) is the cardinality of all extensions over Q_p with degree n in the algebraic closure of Q_p, where p>n. - Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004. Cf. A100976, A100977, A100978 (p-adic extensions).
Let s(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) + a(n-15) - a(n-22) - a(n-26) + ..., then a(n) = s(n) if n is not pentagonal, i.e., n != (3 j^2 +- j)/2 (cf. A001318), and a(n) is instead s(n) - ((-1)^j)*n if n is pentagonal. - Gary W. Adamson, Oct 05 2008 [corrected Apr 27 2012 by William J. Keith based on Ewell and by Andrey Zabolotskiy, Apr 08 2022]
Write n as 2^k * d, where d is odd. Then a(n) is odd if and only if d is a square. - Jon Perry, Nov 08 2012
Also total number of parts in the partitions of n into equal parts. - Omar E. Pol, Jan 16 2013
Note that sigma(3^4) = 11^2. On the other hand, Kanold (1947) shows that the equation sigma(q^(p-1)) = b^p has no solutions b > 2, q prime, p odd prime. - N. J. A. Sloane, Dec 21 2013, based on postings to the Number Theory Mailing List by Vladimir Letsko and Luis H. Gallardo
Limit_{m->infinity} (Sum_{n=1..prime(m)} a(n)) / prime(m)^2 = zeta(2)/2 = Pi^2/12 (A072691). See more at A244583. - Richard R. Forberg, Jan 04 2015
a(n) + A000005(n) is an odd number iff n = 2m^2, m>=1. - Richard R. Forberg, Jan 15 2015
a(n) = a(n+1) for n = 14, 206, 957, 1334, 1364 (A002961). - Zak Seidov, May 03 2016
Equivalent to the Riemann hypothesis: a(n) < H(n) + exp(H(n))*log(H(n)), for all n>1, where H(n) is the n-th harmonic number (Jeffrey Lagarias). See A057641 for more details. - Ilya Gutkovskiy, Jul 05 2016
a(n) is the total number of even parts in the partitions of 2*n into equal parts. More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 18 2019
From Jianing Song, Nov 05 2022: (Start)
a(n) is also the number of order-n subgroups of C_n X C_n, where C_n is the cyclic group of order n. Proof: by the correspondence theorem in the group theory, there is a one-to-one correspondence between the order-n subgroups of C_n X C_n = (Z x Z)/(nZ x nZ) and the index-n subgroups of Z x Z containing nZ x nZ. But an index-n normal subgroup of a (multiplicative) group G contains {g^n : n in G} automatically. The desired result follows from the comment from Naohiro Nomoto above.
The number of subgroups of C_n X C_n that are isomorphic to C_n is A001615(n). (End)

			For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3 + 6 = 12.
Let L = <V,W> be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V,2W>, <2V+W,2W>, <2V,2W+V>. Compare A001615.

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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 116ff.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.
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H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.
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M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.
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G. Pólya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 91, 395.
Robert M. Young, Excursions in Calculus, The Mathematical Association of America, 1992 p. 361.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 20000 terms from N. J. A. Sloane)
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B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
M. Baake and U. Grimm, Quasicrystalline combinatorics
Henry Bottomley, Illustration of initial terms
C. K. Caldwell, The Prime Glossary, sigma function
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012. - _N. J. A. Sloane_, Dec 25 2012
R. L. Duncan, Note on the Divisors of a Number, The American Mathematical Monthly, 68(4), 356-359, (1961).
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal (2004), Vol. 14.1, page 243.
L. Euler, Discovery of a most extraordinary law of numbers, relating to the sum of their divisors
L. Euler, Observatio de summis divisorum
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009.
J. A. Ewell, Recurrences for the sum of divisors, Proc. Amer. Math. Soc. 64 (2) 1977.
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
Johan Gielis and Ilia Tavkhelidze, The general case of cutting of GML surfaces and bodies, arXiv:1904.01414 [math.GM], 2019.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
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Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
Douglas E. Iannucci, On sums of the small divisors of a natural number, arXiv:1910.11835 [math.NT], 2019.
Antti Karttunen, Scheme program for computing this sequence.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math.CO/0503436, 2005.
K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
Walter Nissen, Abundancy : Some Resources
P. Pollack and C. 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 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913.
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. (1992) A48, 500-508. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - _N. J. A. Sloane_, Feb 23 2009
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Eric Weisstein's World of Mathematics, Divisor Function
Wikipedia, Divisor function
Index entries for sequences related to sublattices
Index entries for sequences related to sigma(n)
Index entries for "core" sequences

See A034885, A002093 for records. Bisections give A008438, A062731. Values taken are listed in A007609. A054973 is an inverse function.
For partial sums see A024916.
Row sums of A127093.
sigma_i (i=0..25): A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972, A281959.
Cf. A144736, A158951, A158902, A174740, A147843, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices), A039653, A088580, A074400, A083728, A006352, A002659, A083238, A000593, A050449, A050452, A051731, A027748, A124010, A069192, A057641, A001318.
Cf. A009194, A082062 (gcd(a(n),n) and its largest prime factor), A179931, A192795 (gcd(a(n),A001157(n)) and largest prime factor).
Cf. also A034448 (sum of unitary divisors).
Cf. A007955 (products of divisors).
Cf. A144613, A144614, A144615, A146076.
A001227, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
Cf. A001221, A054533.

A000203:=List([1..10^2],n->Sigma(n)); # Muniru A Asiru, Oct 01 2017

a000203 n = product $ zipWith (\p e -> (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, May 07 2012

Magma
```
[SumOfDivisors(n): n in [1..70]];
```

[DivisorSigma(1,n): n in [1..70]]; // Bruno Berselli, Sep 09 2015

with(numtheory): A000203 := n->sigma(n); seq(A000203(n), n=1..100);

Table[ DivisorSigma[1, n], {n, 100}]
a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* Michael Somos, Apr 25 2013 *)

makelist(divsum(n),n,1,1000); /* Emanuele Munarini, Mar 26 2011 */

numlib::sigma(n)$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI
```
{a(n) = if( n<1, 0, sigma(n))};
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) /(1 - p*X))[n])};

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k)^2, x * O(x^n)), n))}; /* Michael Somos, Jan 29 2005 */

max_n = 30; ser = - sum(k=1,max_n,log(1-x^k)); a(n) = polcoeff(ser,n)*n \\ Gottfried Helms, Aug 10 2009

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jan 03 2021

from math import prod
from sympy import factorint
def a(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024
(APL, Dyalog dialect) A000203 ← +/{ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð} ⍝ Antti Karttunen, Feb 20 2024

[sigma(n, 1) for n in range(1, 71)]  # Zerinvary Lajos, Jun 04 2009

(definec (A000203 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (/ (- (expt p (+ 1 e)) 1) (- p 1)) (A000203 (A028234 n)))))) ;; Uses macro definec from http://oeis.org/wiki/Memoization#Scheme - Antti Karttunen, Nov 25 2017

(define (A000203 n) (let ((r (sqrt n))) (let loop ((i (inexact->exact (floor r))) (s (if (integer? r) (- r) 0))) (cond ((zero? i) s) ((zero? (modulo n i)) (loop (- i 1) (+ s i (/ n i)))) (else (loop (- i 1) s)))))) ;; (Stand-alone program) - Antti Karttunen, Feb 20 2024

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson, Aug 01 2001
For the following bounds and many others, see Mitrinovic et al. - N. J. A. Sloane, Oct 02 2017
If n is composite, a(n) > n + sqrt(n).
a(n) < n*sqrt(n) for all n.
a(n) < (6/Pi^2)*n^(3/2) for n > 12.
G.f.: -x*deriv(eta(x))/eta(x) where eta(x) = Product_{n>=1} (1-x^n). - Joerg Arndt, Mar 14 2010
L.g.f.: -log(Product_{j>=1} (1-x^j)) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Dirichlet convolution of phi(n) and tau(n), i.e., a(n) = sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.
a(n) is odd iff n is a square or twice a square. - Robert G. Wilson v, Oct 03 2001
a(n) = a(n*prime(n)) - prime(n)*a(n). - Labos Elemer, Aug 14 2003 (Clarified by Omar E. Pol, Apr 27 2016)
a(n) = n*A000041(n) - Sum_{i=1..n-1} a(i)*A000041(n-i). - Jon Perry, Sep 11 2003
a(n) = -A010815(n)*n - Sum_{k=1..n-1} A010815(k)*a(n-k). - Reinhard Zumkeller, Nov 30 2003
a(n) = f(n, 1, 1, 1), where f(n, i, x, s) = if n = 1 then s*x else if p(i)|n then f(n/p(i), i, 1+p(i)*x, s) else f(n, i+1, 1, s*x) with p(i) = i-th prime (A000040). - Reinhard Zumkeller, Nov 17 2004
Recurrence: n^2*(n-1)*a(n) = 12*Sum_{k=1..n-1} (5*k*(n-k) - n^2)*a(k)*a(n-k), if n>1. - Dominique Giard (dominique.giard(AT)gmail.com), Jan 11 2005
G.f.: Sum_{k>0} k * x^k / (1 - x^k) = Sum_{k>0} x^k / (1 - x^k)^2. Dirichlet g.f.: zeta(s)*zeta(s-1). - Michael Somos, Apr 05 2003. See the Hardy-Wright reference, p. 312. first equation, and p. 250, Theorem 290. - Wolfdieter Lang, Dec 09 2016
For odd n, a(n) = A000593(n). For even n, a(n) = A000593(n) + A074400(n/2). - Jonathan Vos Post, Mar 26 2006
Equals the inverse Moebius transform of the natural numbers. Equals row sums of A127093. - Gary W. Adamson, May 20 2007
A127093 * [1/1, 1/2, 1/3, ...] = [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, ...]. Row sums of triangle A135539. - Gary W. Adamson, Oct 31 2007
a(n) = A054785(2*n) - A000593(2*n). - Reinhard Zumkeller, Apr 23 2008
a(n) = n*Sum_{k=1..n} A060642(n,k)/k*(-1)^(k+1). - Vladimir Kruchinin, Aug 10 2010
Dirichlet convolution of A037213 and A034448. - R. J. Mathar, Apr 13 2011
G.f.: A(x) = x/(1-x)*(1 - 2*x*(1-x)/(G(0) - 2*x^2 + 2*x)); G(k) = -2*x - 1 - (1+x)*k + (2*k+3)*(x^(k+2)) - x*(k+1)*(k+3)*((-1 + (x^(k+2)))^2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2011
a(n) = A001065(n) + n. - Mats Granvik, May 20 2012
a(n) = A006128(n) - A220477(n). - Omar E. Pol, Jan 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A196020(n,k). - conjectured by Omar E. Pol, Feb 02 2013, and proved by Max Alekseyev, Nov 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A000330(k)*A000716(n-A000217(k)). - Mircea Merca, Mar 05 2014
a(n) = A240698(n, A000005(n)). - Reinhard Zumkeller, Apr 10 2014
a(n) = Sum_{d^2|n} A001615(n/d^2) = Sum_{d^3|n} A254981(n/d^3). - Álvar Ibeas, Mar 06 2015
a(3*n) = A144613(n). a(3*n + 1) = A144614(n). a(3*n + 2) = A144615(n). - Michael Somos, Jul 19 2015
a(n) = Sum{i=1..n} Sum{j=1..i} cos((2*Pi*n*j)/i). - Michel Lagneau, Oct 14 2015
a(n) = A000593(n) + A146076(n). - Omar E. Pol, Apr 05 2016
a(n) = A065475(n) + A048050(n). - Omar E. Pol, Nov 28 2016
a(n) = (Pi^2*n/6)*Sum_{q>=1} c_q(n)/q^2, with the Ramanujan sums c_q(n) given in A054533 as a c_n(k) table. See the Hardy reference, p. 141, or Hardy-Wright, Theorem 293, p. 251. - Wolfdieter Lang, Jan 06 2017
G.f. also (1 - E_2(q))/24, with the g.f. E_2 of A006352. See e.g., Hardy, p. 166, eq. (10.5.5). - Wolfdieter Lang, Jan 31 2017
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(n) + A162296(n).
a(n) = A092261(n) * A295294(n). [This can be further expanded, see comment in A291750.] (End)
a(n) = A000593(n) * A038712(n). - Ivan N. Ianakiev and Omar E. Pol, Nov 26 2017
a(n) = Sum_{q=1..n} c_q(n) * floor(n/q), where c_q(n) is the Ramanujan's sum function given in A054533. - Daniel Suteu, Jun 14 2018
a(n) = Sum_{k=1..n} gcd(n, k) / phi(n / gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 21 2018
a(n) = (2^(1 + (A000005(n) - A001227(n))/(A000005(n) - A183063(n))) - 1)*A000593(n) = (2^(1 + (A183063(n)/A001227(n))) - 1)*A000593(n). - Omar E. Pol, Nov 03 2018
a(n) = Sum_{i=1..n} tau(gcd(n, i)). - Ridouane Oudra, Oct 15 2019
From Peter Bala, Jan 19 2021: (Start)
G.f.: A(x) = Sum_{n >= 1} x^(n^2)*(x^n + n*(1 - x^(2*n)))/(1 - x^n)^2 - differentiate equation 5 in Arndt w.r.t. x, and set x = 1.
A(x) = F(x) + G(x), where F(x) is the g.f. of A079667 and G(x) is the g.f. of A117004. (End)
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
With the convention that a(n) = 0 for n <= 0 we have the recurrence a(n) = t(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where t(n) = (-1)^(m+1)*(2*m+1)*n/3 if n = m*(m + 1)/2, with m positive, is a triangular number else t(n) = 0. For example, n = 10 = (4*5)/2 is a triangular number, t(10) = -30, and so a(10) = -30 + 3*a(9) - 5*a(7) + 7*a(4) = -30 + 39 - 40 + 49 = 18. - Peter Bala, Apr 06 2022
Recurrence: a(p^x) = p*a(p^(x-1)) + 1, if p is prime and for any integer x. E.g., a(5^3) = 5*a(5^2) + 1 = 5*31 + 1 = 156. - Jules Beauchamp, Nov 11 2022
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = A319462. - Vaclav Kotesovec, May 07 2023
a(n) < (7n*A001221(n) + 10*n)/6 [Duncan, 1961] (see Duncan and Tattersall). - Stefano Spezia, Jul 13 2025

A005153 Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252

Offset: 1

N. J. A. Sloane

Equivalently, positive integers m such that every number k <= m is a sum of distinct divisors of m.
2^r is a member for all r as every number < = sigma(2^r) = 2^(r+1)-1 is a sum of a distinct subset of divisors {1, 2, 2^2, ..., 2^m}. - Amarnath Murthy, Apr 23 2004
Also, numbers m such that A030057(m) > m. This is a consequence of the following theorem (due to Stewart), found at the McLeman link: An integer m >= 2 with factorization Product_{i=1..k} p_i^e_i with the p_i in ascending order is practical if and only if p_1 = 2 and, for 1 < i <= k, p_i <= sigma(Product_{j < i} p_j^e_j) + 1. - Franklin T. Adams-Watters, Nov 09 2006
Practical numbers first appear in Srinivasan's short paper, which contains terms up to 200. Let m be a practical number. He states that (1) if m>2, m is a multiple of 4 or 6; (2) sigma(m) >= 2*m-1 (A103288); and (3) 2^t*m is practical. He also states that highly composite numbers (A002182), perfect numbers (A000396), and primorial numbers (A002110) are practical. - T. D. Noe, Apr 02 2010
Conjecture: The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing to the limit 1. - Zhi-Wei Sun, Jan 12 2013
Conjecture: For any positive rational number r, there are finitely many pairwise distinct practical numbers q(1)..q(k) such that r = Sum_{j=1..k} 1/q(j). For example, 2 = 1/1 + 1/2 + 1/4 + 1/6 + 1/12 with 1, 2, 4, 6 and 12 all practical, and 10/11 = 1/2 + 1/4 + 1/8 + 1/48 + 1/132 + 1/176 with 2, 4, 8, 48, 132 and 176 all practical. - Zhi-Wei Sun, Sep 12 2015
Analogous with the {1 union primes} (A008578), practical numbers form a complete sequence. This is because it contains all powers of 2 as a subsequence. - Frank M Jackson, Jun 21 2016
Sun's 2015 conjecture on the existence of Egyptian fractions with practical denominators for any positive rational number is true. See the link "Egyptian fractions with practical denominators". - David Eppstein, Nov 20 2016
Conjecture: if all divisors of m are 1 = d_1 < d_2 < ... < d_k = m, then m is practical if and only if d_(i+1)/d_i <= 2 for 1 <= i <= k-1. - Jianing Song, Jul 18 2018
The above conjecture is incorrect. The smallest counterexample is 78 (for which one of these quotients is 13/6; see A174973). m is practical if and only if the divisors of m form a complete subsequence. See Wikipedia links. - Frank M Jackson, Jul 25 2018
Reply to the comment above: Yes, and now I can show the opposite: The largest value of d_(i+1)/d_i is not bounded for practical numbers. Note that sigma(n)/n is not bounded for primorials, and primorials are practical numbers. For any constant c >= 2, let k be a practical number such that sigma(k)/k > 2c. By Bertrand's postulate there exists some prime p such that c*k < p < 2c*k < sigma(k), so k*p is a practical number with consecutive divisors k and p where p/k > c. For example, for k = 78 we have 13/6 > 2, and for 97380 we have 541/180 > 3. - Jianing Song, Jan 05 2019
Erdős (1950) and Erdős and Loxton (1979) proved that the asymptotic density of practical numbers is 0. - Amiram Eldar, Feb 13 2021
Let P(x) denote the number of practical numbers up to x. P(x) has order of magnitude x/log(x) (see Saias 1997). Moreover, we have P(x) = c*x/log(x) + O(x/(log(x))^2), where c = 1.33607... (see Weingartner 2015, 2020 and Remark 1 of Pomerance & Weingartner 2021). As a result, a(n) = k*n*log(n*log(n)) + O(n), where k = 1/c = 0.74846... - Andreas Weingartner, Jun 26 2021
From Hal M. Switkay, Dec 22 2022: (Start)
Every number of least prime signature (A025487) is practical, thereby including two classes of number mentioned in Noe's comment. This follows from Stewart's characterization of practical numbers, mentioned in Adams-Watters's comment, combined with Bertrand's postulate (there is a prime between every natural number and its double, inclusive).
Also, the first condition in Stewart's characterization (p_1 = 2) is equivalent to the second condition with index i = 1, given that an empty product is equal to 1. (End)
Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number. Note that this conjecture occupies the space between the unproven Goldbach conjecture and the theorem that every even number, beginning with 2, is the sum of two practical numbers (Melfi's 1996 proof of Margenstern's conjecture). - Hal M. Switkay, Jan 28 2023

H. Heller, Mathematical Buds, Vol. 1, Chap. 2, pp. 10-22, Mu Alpha Theta OK, 1978.
Malcolm R. Heyworth, More on Panarithmic Numbers, New Zealand Math. Mag., Vol. 17 (1980), pp. 28-34 [ ISSN 0549-0510 ].
Ross 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).
A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 146-147.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Wayne Dymacek, Letter to N. J. A. Sloane, Jun 15 1978.
David Eppstein, Egyptian fractions with practical denominators, Nov 20, 2016
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
Paul Erdős, On a Diophantine equation (in Hungarian, with Russian and English summaries), Mat. Lapok, Vol. 1 (1950), pp. 192-210.
Paul Erdős and J. H. Loxton, Some problems in partitio numerorum, Journal of the Australian Mathematical Society, Vol. 27, No. 3 (1979), pp. 319-331.
James Grimes and Brady Haran, Practical Numbers, Numberphile video (2023).
Harvey J. Hindin, Quasipractical numbers, IEEE Communications Magazine, Vol. 18, No. 2 (March 1980), pp. 41-45.
Paolo Leonetti and Carlo Sanna, Practical numbers among the binomial coefficients, Journal of Number Theory, Vol. 207 (2020), pp. 145-155; arXiv preprint, arXiv:1905.12023 [math.NT], 2019.
Maurice Margenstern, Sur les nombres pratiques, (in French), Groupe d'étude en théorie analytique des nombres, 1 (1984-1985), Exposé No. 21, 13 p.
Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, Journal of Number Theory, Volume 37, Issue 1 (January 1991), pp. 1-36.
C. McLeman, Practical number, PlanetMath.org.
Giuseppe Melfi, On two conjectures about practical numbers, J. Number Theory, Vol. 56, No. 1 (1996), pp. 205-210 [MR96i:11106].
Giuseppe Melfi, On certain positive integer sequences, arXiv:0404555 [math.NT], 2004.
Giuseppe Melfi, A survey on practical numbers, Rend. Sem. Mat. Univ. Politec. Torino, Vol. 53, No. 4 (1995), pp. 347-359.
Giuseppe Melfi, Practical Numbers (old link).
Paul Pollack and Lola Thompson, Practical pretenders, arXiv:1201.3168 [math.NT], Jan 16 2012.
Carl Pomerance, Lola Thompson and Andreas Weingartner, On integers n for which X^n-1 has a divisor of every degree, arXiv:1511.03357 [math.NT], 2015.
Carl Pomerance and Andreas Weingartner, On primes and practical numbers, Ramanujan J. (2021); arXiv preprint, arXiv:2007.11062 [math.NT], 2020.
Eric Saias, Entiers à diviseurs denses 1, J. Number Theory, Vol. 62, No. 1 (1997), pp. 163-191; uses this definition.
Carlo Sanna, Practical central binomial coefficients, arXiv:2004.05376 [math.NT], 2020.
Sai Teja Somu, Ting Hon Stanford Li, and Andrzej Kukla, On Some Results on Practical Numbers, INTEGERS, Volume 23, A68, 2023 [MR4643065].
Sai Teja Somu and Duc Van Khanh Tran, On Sums of Practical Numbers and Polygonal Numbers, arXiv:2403.13533 [math.NT], 2024.
A. K. Srinivasan, Practical numbers, Current Science, 17 (1948), 179-180.
B. M. Stewart, Sums of distinct divisors, Amer. J. Math., Vol. 76, No. 4 (1954), pp. 779-785 [MR64800]
Zhi-Wei Sun, A conjecture on unit fractions involving primes, preprint, 2015.
Peter Taylor, Table of n, a(n) for n = 1..1000000.
Andreas Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics, Vol. 66, No. 2 (2015), pp. 743-758; arXiv preprint, arXiv:1405.2585 [math.NT], 2014-2015.
Andreas Weingartner, The constant factor in the asymptotic for practical numbers, Int. J. Number Theory, 16 (2020), no. 3, 629-638; arXiv preprint, arXiv:1906.07819 [math.NT], 2019.
Andreas Weingartner, Uniform distribution of alpha*n modulo one for a family of integer sequences, arXiv:2303.16819 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Practical Number.
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]
Wikipedia, Practical number.
Robert G. Wilson v, Letter to N. J. A. Sloane, date unknown.

Subsequence of A103288.
Cf. A002093, A007620 (second definition), A030057, A033630, A119348, A174533, A174973.
Cf. A027750.

a005153 n = a005153_list !! (n-1)
a005153_list = filter (\x -> all (p $ a027750_row x) [1..x]) [1..]
   where p _  0 = True
         p [] _ = False
         p ds'@(d:ds) m = d <= m && (p ds (m - d) || p ds m)
-- Reinhard Zumkeller, Feb 23 2014, Oct 27 2011

isA005153 := proc(n)
    local ifs,pprod,p,i ;
    if n = 1 then
        return true;
    elif type(n,'odd') then
        return false ;
    end if;
    # not using ifactors here directly because no guarantee primes are sorted...
    ifs := ifactors(n)[2] ;
    pprod := 1;
    for p in sort(numtheory[factorset](n) ) do
        for i in ifs do
            if op(1,i) = p then
                if p > 2 and p > 1+numtheory[sigma](pprod) then
                    return false ;
                end if;
                pprod := pprod*p^op(2,i) ;
            end if;
        end do:
    end do:
    return true ;
end proc:
for n from 1 to 300 do
    if isA005153(n)  then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 07 2023

PracticalQ[n_] := Module[{f,p,e,prod=1,ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p,e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1,prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i,Length[p]}]; ok]]]; Select[Range[200], PracticalQ] (* T. D. Noe, Apr 02 2010 *)

is_A005153(n)=bittest(n,0) && return(n==1); my(P=1); n && !for(i=2,#n=factor(n)~,n[1,i]>1+(P*=sigma(n[1,i-1]^n[2,i-1])) && return) \\ M. F. Hasler, Jan 13 2013

from sympy import factorint
def is_A005153(n):
    if n & 1: return n == 1
    f = factorint(n) ; P = (2 << f.pop(2)) - 1
    for p in f: # factorint must have prime factors in increasing order
        if p > 1 + P: return
        P *= p**(f[p]+1)//(p-1)
    return True # M. F. Hasler, Jan 02 2023

from sympy import divisors;from more_itertools import powerset
[i for i in range(1,253) if (lambda x:len(set(map(sum,powerset(x))))>sum(x))(divisors(i))] # Nicholas Stefan Georgescu, May 20 2023

Weingartner proves that a(n) ~ k*n log n, strengthening an earlier result of Saias. In particular, a(n) = k*n log n + O(n log log n). - Charles R Greathouse IV, May 10 2013

More precisely, a(n) = k*n*log(n*log(n)) + O(n), where k = 0.74846... (see comments). - Andreas Weingartner, Jun 26 2021

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
Erroneous comment removed by T. D. Noe, Nov 14 2010
Definition changed to exclude n = 0 explicitly by M. F. Hasler, Jan 19 2013

A004394 Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600

Offset: 1

Matthew Conroy

Matthew Conroy points out that these are different from the highly composite numbers - see A002182. Jul 10 1996
With respect to the comment above, neither sequence is subsequence of the other. - Ivan N. Ianakiev, Feb 11 2020
Also n such that sigma_{-1}(n) > sigma_{-1}(m) for all m < n, where sigma_{-1}(n) is the sum of the reciprocals of the divisors of n. - Matthew Vandermast, Jun 09 2004
Ramanujan (1997, Section 59; written in 1915) called these numbers "generalized highly composite." Alaoglu and Erdős (1944) changed the terminology to "superabundant." - Jonathan Sondow, Jul 11 2011
Alaoglu and Erdős show that: (1) n is superabundant => n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2 >= e_3 >= ... >= e_p (and e_p is 1 unless n=4 or n=36); (2) if q < r are primes, then | e_r - floor(e_q*log(q)/log(r)) | <= 1; (3) q^{e_q} < 2^{e_2+2} for primes q, 2 < q <= p. - Keith Briggs, Apr 26 2005
It follows from Alaoglu and Erdős finding 1 (above) that, for n > 7, a(n) is a Zumkeller Number (A083207); for details, see Proposition 9 and Corollary 5 at Rao/Peng link (below). - Ivan N. Ianakiev, Feb 11 2020
See A166735 for superabundant numbers that are not highly composite, and A189228 for superabundant numbers that are not colossally abundant.
Pillai called these numbers "highly abundant numbers of the 1st order". - Amiram Eldar, Jun 30 2019

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad. Publ., 2002 (see pp. 19-21).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

D. Kilminster, Table of n, a(n) for n = 1..2000 (extends to n = 8436 in the comments; first 500 terms from T. D. Noe)
A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, Amer. Math. Monthly, 116 (2009), 273-275.
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.
Christian Axler, Inequalities involving the primorial counting function, arXiv:2406.04018 [math.NT], 2024. See p. 12.
Yu. Bilu, P. Habegger, and L. Kühne, Effective bounds for singular units, arXiv:1805.07167 [math.NT], 2018.
Benjamin Braun and Brian Davis, Antichain Simplices, arXiv:1901.01417 [math.CO], 2019.
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Tibor Burdette and Ian Stewart, Counterexamples to a Conjecture by Alaoglu and Erdős, arXiv:2009.03306 [math.NT], 2020.
Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, INTEGERS 11 (2011), #A33.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
François Clément and Stefan Steinerberger, On the largest singular vector of the Redheffer matrix, arXiv:2502.09489 [math.NT], 2025. See p. 2.
P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants (Text in French), Bulletin de la S. M. F., tome 103 (1975), pp. 65-90.
F. Jokar, On k-layered numbers and some labeling related to k-layered numbers, arXiv:2003.11309 [math.NT], 2020.
Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
A. Morkotun, On the increase of Gronwall function value at the multiplication of its argument by a prime, arXiv preprint arXiv:1307.0083 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Superabundant numbers, their subsequences and the Riemann hypothesis, arXiv preprint arXiv:1211.2147 [math.NT], 2012.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Walter Nissen, Abundancy : Some Resources.
T. D. Noe, First 500 superabundant numbers.
T. D. Noe, First 1000000 superabundant numbers (21 MB, zipped).
S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Superabundant Number.
Wikipedia, Superabundant number.

Almost the same as A077006.
The colossally abundant numbers A004490 are a subsequence, as are A023199.
Subsequence of A025487; apart from a(3) = 4 and a(7) = 36, a subsequence of A102750.
Cf. A000203, A002093, A002182.
Cf. A112974 (number of superabundant numbers between colossally abundant numbers).
Cf. A091901 (Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality), A192884 (non-superabundant and the reverse of Robin's inequality).

a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
(* Second program: convert all 8436 terms in b-file into a list of terms: *)
f[w_] := Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[w, ?(! DigitQ@ # &)], StringCases[w, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[w, (x : DigitCharacter ..) ~~ "!" :> x]}; Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ] (* _Michael De Vlieger, May 08 2018 *)

print1(r=1);forstep(n=2,1e6,2,t=sigma(n,-1);if(t>r,r=t;print1(", "n))) \\ Charles R Greathouse IV, Jul 19 2011

a(n+1) <= 2*a(n). - A.H.M. Smeets, Jul 10 2021

Name edited by Peter Munn, Mar 13 2019

All superabundant numbers (A004394), colossally abundant numbers (A004490), highly composite numbers (A002182) and superior highly composite numbers (A002201) are Harshad numbers. However, this is not true of the highly abundant numbers (A002093) and there are 32 exceptions in the 394 highly abundant numbers less than 50 million.

The previous comment is erroneous. The first superabundant number that is not a Harshad number is A004394(105) = 149602080797769600. The first highly composite number that is not a Harshad number is A002182(61) = 245044800. For all exceptions I found, the sum of digits is a power of 3. Although the first 60000 terms of the colossally abundant numbers and the superior highly composite numbers are Harshad numbers, I am not aware of a proof that all terms are Harshad numbers. There may be large counterexamples. [T. D. Noe, Oct 27 2009]

Define "largely abundant numbers" to be numbers k such that sigma(k) >= sigma(j) for all j < k. This sequence gives all the largely abundant numbers that are not highly abundant numbers.
Analogous to A244353 as A002093 is analogous to A002182.
No more terms below 10^10.
a(22) > 10^13. - Giovanni Resta, Jul 02 2019

cp's OEIS Frontend

A128699 Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.

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A128702 Highly abundant numbers (A002093) that are not Harshad numbers (A005349).

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A228944 Number of ways to write highly composite numbers (A002182(n)) as the difference of two highly abundant numbers (A002093), both <= 2*A002182(n).

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A349608 a(n) is the number of divisors of the n-th highly abundant number (A002093).

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A181310 Highly abundant numbers (A002093) whose largest prime factor has power greater than 1.

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A228942 Number of decompositions of highly composite numbers (A002182) into unordered sums of two highly abundant numbers (A002093).

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A308574 Numbers between a pair of consecutive highly abundant numbers (A002093) having the same sum of divisors as the lesser one.

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A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

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