A134675 -id:A134675 - cp's OEIS frontend

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106

Offset: 1

N. J. A. Sloane, R. K. Guy

Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part. - Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples. - Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest. - Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 23 2019
Fixed points are in A000396. - Alois P. Heinz, Mar 10 2024

			x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 84-44 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 -> 14288 -> 15472 -> 14536 -> 14264 -> 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8. - _Omar E. Pol_, Nov 24 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 75-92; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
Carl Pomerance, The first function and its iterates, pp. 125-138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
H. J. J. te Riele, Perfect numbers and aliquot sequences, pp. 77-94 in J. van de Lune, ed., Studieweek "Getaltheorie en Computers", published by Math. Centrum, Amsterdam, Sept. 1980.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 91.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [alternative scanned copy].
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Henry Bottomley, Illustration of initial terms
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
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., to appear (2014).
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series: Abundant, deficient, perfect
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences

Least inverse: A070015, A359132.
Values taken: A078923, values not taken: A005114.
Records: A034090, A034091.
First differences: A053246, partial sums: A153485.
a(n) = n - A033879(n) = n + A033880(n). - Omar E. Pol, Dec 30 2013
Row sums of A141846 and of A176891. - Gary W. Adamson, May 02 2010
Row sums of A176079. - Mats Granvik, May 20 2012
Alternating row sums of A231347. - Omar E. Pol, Jan 02 2014
a(n) = sum (A027751(n,k): k = 1..A000005(n)-1). - Reinhard Zumkeller, Apr 05 2013
For n > 1: a(n) = A240698(n,A000005(n)-1). - Reinhard Zumkeller, Apr 10 2014
A134675(n) = A007434(n) + a(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
Cf. A032741, A000203, A048050, A000593, A027750.
Cf. A051953, A051731.
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
Cf. A007956 (products of proper divisors).
Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).

a001065 n = a000203 n - n  -- Reinhard Zumkeller, Sep 15 2011

[SumOfDivisors(n)-n: n in [1..100]]; // Vincenzo Librandi, May 06 2015

A001065 := proc(n)
    numtheory[sigma](n)-n ;
end proc:
seq( A001065(n),n=1..100) ;

Table[ Plus @@ Select[ Divisors[ n ], #Zak Seidov, Sep 10 2009 *)
Table[DivisorSigma[1, n] - n, {n, 1, 80}] (* Jean-François Alcover, Apr 25 2013 *)
Array[Plus @@ Most@ Divisors@# &, 80] (* Robert G. Wilson v, Dec 24 2017 *)

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

{a(n) = if( n==0, 0, sigma(n) - n)} /* Michael Somos, Sep 20 2011 */

from sympy import divisor_sigma
def A001065(n): return divisor_sigma(n)-n # Chai Wah Wu, Nov 04 2022

[sigma(n, 1)-n for n in range(1, 81)] # Stefano Spezia, Jul 14 2025

G.f.: Sum_{k>0} k * x^(2*k)/(1 - x^k). - Michael Somos, Jul 05 2006
a(n) = sigma(n) - n = A000203(n) - n. - Lekraj Beedassy, Jun 02 2005
a(n) = A155085(-n). - Michael Somos, Sep 20 2011
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6. - Gary W. Adamson, Jul 11 2008
a(n) = A006128(n) - A220477(n) - n. - Omar E. Pol Jan 17 2013
a(n) = Sum_{i=1..floor(n/2)} i*(1-ceiling(frac(n/i))). - Wesley Ivan Hurt, Oct 25 2013
Dirichlet g.f.: zeta(s-1)*(zeta(s) - 1). - Ilya Gutkovskiy, Aug 07 2016
a(n) = 1 + A048050(n), n > 1. - R. J. Mathar, Mar 13 2018
Erdős (Elem. Math. 28 (1973), 83-86) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1. - N. J. A. Sloane, Dec 21 2019
G.f.: Sum_{k >= 2} x^k/(1 - x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1 - z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.) - Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2) - (n + 1)*q^(2*n+1) - (n - 1)*q^(n+1) + n)/((1 - q^n)*(1 - q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get -t/(1 - t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.) - Peter Bala, Jan 22 2021
a(n) = Sum_{d|n} d * (1 - [n = d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
a(n) = Sum_{i=1..n} ((n-1) mod i) - (n mod i). [See also A176079.] - José de Jesús Camacho Medina, Feb 23 2021

A007434 Jordan function J_2(n) (a generalization of phi(n)).

Original entry on oeis.org

1, 3, 8, 12, 24, 24, 48, 48, 72, 72, 120, 96, 168, 144, 192, 192, 288, 216, 360, 288, 384, 360, 528, 384, 600, 504, 648, 576, 840, 576, 960, 768, 960, 864, 1152, 864, 1368, 1080, 1344, 1152, 1680, 1152, 1848, 1440, 1728, 1584, 2208, 1536

Offset: 1

N. J. A. Sloane

Number of points in the bicyclic group Z/mZ X Z/mZ whose order is exactly m. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Mar 14 2006
Number of irreducible fractions among {(u+v*i)/n : 1 <= u, v <= n} with i = sqrt(-1), where a fraction (u+v*i)/n is called irreducible if and only if gcd(u, v, n) = 1. - Reinhard Zumkeller, Aug 20 2005
The weight of the n-th polynomial for the analog of cyclotomic polynomials for elliptic divisibility sequences. That is, let the weight of b1 = 1, b2 = 3, b3 = 8, b4 = 12 and let e1 = b1, e2 = b2*b1, e3 = b3*b1, e4 = b2*b4*b1, e5 = (b2^4*b4 - b3^3)*b1 = b5*e1, and so on, be an elliptic divisibility sequence. Then weight of e2 = 4, e3 = 9, e4 = 16, e5 = 25, where weight of en is n^2 in general, while weight of bn is a(n). - Michael Somos, Aug 12 2008
J_2(n) divides J_{2k}(n). J_2(n) gives the number of 2-tuples (x1,x2), such that 1 <= x1, x2 <= n and gcd(x1, x2, n) = 1. - Enrique Pérez Herrero, Mar 05 2011
From Jianing Song, Apr 06 2019: (Start)
Let k be any quadratic field such that all prime factors of n are inert in k, O_k be the corresponding ring of integers and G(n) = (O_k/nO_k)* be the multiplicative group of integers in O_k modulo n, then a(n) is the number of elements in G(n). The exponent of G(n) is A306933(n). [Equivalently, G(p^e) can be defined as (Z_{p^2}/p^eZ_{p^2})*, where Z_{p^2} is the ring of integers of the field Q_{p^2} (with a unique maximal ideal pZ_{p^2}), and Q_{p^2} is the unique unramified quadratic extension of the p-adic field Q_p. For the group structure of G(p^e), see A306933. - Jianing Song, Jun 19 2025]
For n >= 5, a(n) is divisible by 24. (End)
The Del Centina article on page 106 mentions a formula by Halphen denoted by phi(n)T(n). - Michael Somos, Feb 05 2021

			a(4) = 12 because the divisors of 4 being 1, 2, 4, we find that phi(1)*phi(4/1)*(4/1) = 8, phi(2)*phi(4/2)*(4/2) = 2, phi(4)*phi(4/4)*(4/4) = 2 and 8 + 2 + 2 = 12.
G.f. = x + 3*x^2 + 8*x^3 + 12*x^4 + 24*x^5 + 24*x^6 + 48*x^7 + 48*x^8 + 72*x^9 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
A. Del Centina, Poncelet's porism: a long story of renewed discoveries, I, Hist. Exact Sci. (2016), v. 70, p. 106.
L. E. Dickson (1919, repr. 1971). History of the Theory of Numbers I. Chelsea. p. 147.
P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, NY, USA, 1986.
G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Section 6, Problem 64.
M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. 206. Springer-Verlag. p. 11.
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
Sukumar Das Adhikari and A. Sankaranarayanan, On an error term related to the Jordan totient function Jk(n), Journal of Number Theory Volume 34, Issue 2, February 1990, Pages 178-188.
Dorin Andrica and Mihai Piticari, On Some Extensions Of Jordan's Arithmetic Functions, Proceedings of the International Conference on Theory and Applications of Mathematics and Informatics - ICTAMI 2003, Alba Iulia.
E. Artal Bartolo, P. D. González Pérez, M. González Villa, and E. León-Cardenal, On a suspension formula for Denef-Loeser zeta functions, arXiv:2502.05022 [math.AG], 2025. See p. 19.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 32.
F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
H. Li and T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 38.
MathOverflow, Averages of euler-phi function and similar.
Carl-Fredrik Nyberg-Brodda, On congruence subgroups of SL_2(Z[1/p]) generated by two parabolic elements, arXiv:2312.11258 [math.GR], 2023.
Nittiya Pabhapote and Vichian Laohakosol, Combinatorial Aspects of the Generalized Euler's Totient, International Journal of Mathematics and Mathematical Sciences, Volume 2010 (2010), Article ID 648165, 15 p.
Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.
N. J. A. Sloane, Transforms.
S. Thajoddin and S. Vangipuram, A Note On Jordan's Totient Function, Indian J. Pure Appl. Math, 1988.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
Wikipedia, Jordan's totient function.
Index to divisibility sequences

Cf. A000056, A000290, A001065, A001615, A027750, A046970, A115000, A134675.
Cf. A059379 and A059380 (triangle of values of J_k(n)).
Cf. A000010 (J_1), this sequence (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).
Cf. A002117, A088453, A301875, A301876, A321879 (partial sums).

a007434 n = sum $ zipWith3 (\x y z -> x * y * z)
                  tdivs (reverse tdivs) (reverse divs)
                  where divs = a027750_row n;  tdivs = map a000010 divs
-- Reinhard Zumkeller, Nov 24 2012

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 2)
A007434 := proc(n)
    add(d^2*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Nov 03 2015

jordanTotient[n_, k_:1] := DivisorSum[n, #^k*MoebiusMu[n/#] &] /; (n > 0) && IntegerQ[n]; Table[jordanTotient[n, 2], {n, 48}] (* Enrique Pérez Herrero, Sep 14 2010 *)
a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ n/d], {d, Divisors @ n}]]; (* Michael Somos, Jan 11 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], n^2 (Times @@ ((1 - 1/#[[1]]^2) & /@ FactorInteger @ n))]; (* Michael Somos, Jan 11 2014 *)
jordanTotient[n_Integer?Positive, r_:1] := DirichletConvolve[MoebiusMu[K], K^r, K, n]; Table[jordanTotient[n, 2], {n, 48}] (* Jan Mangaldan, Jun 03 2016 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d^2 * moebius(n / d)))}; /* Michael Somos, Mar 20 2004 */

{a(n) = if( n<1, 0, direuler( p=2, n, (1 - X) / (1 - X*p^2))[n])}; /* Michael Somos, Jan 11 2014 */

seq(n) = dirmul(vector(n,k,k^2), vector(n,k,moebius(k)));
seq(48)  \\ Gheorghe Coserea, May 11 2016

jordan(n,k)=my(a=n^k);fordiv(n,i,if(isprime(i),a*=(1-1/(i^k))));a  \\ Roderick MacPhee, May 05 2017

from math import prod
from sympy import factorint
def A007434(n): return prod(p**(e-1<<1)*(p**2-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

Moebius transform of squares.
Multiplicative with a(p^e) = p^(2e) - p^(2e-2). - Vladeta Jovovic, Jul 26 2001
a(n) = Sum_{d|n} d^2 * mu(n/d). - Benoit Cloitre, Apr 05 2002
a(n) = n^2 * Product_{p|n} (1-1/p^2). - Tom Edgar, Jan 07 2015
a(n) = Sum_{d|n} phi(d)*phi(n/d)*n/d; Sum_{d|n} a(d) = n^2. - Reinhard Zumkeller, Aug 20 2005
Dirichlet generating function: zeta(s-2)/zeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Dirichlet inverse of A046970. - Michael Somos, Jan 11 2014
a(n) = a(n^2)/n^2. - Enrique Pérez Herrero, Sep 14 2010
a(n) = A000010(n) * A001615(n).
If n > 1, then 1 > a(n)/n^2 > 1/zeta(2). - Enrique Pérez Herrero, Jul 14 2011
a(n) = Sum_{d|n} phi(n^2/d)*mu(d)^2. - Enrique Pérez Herrero, Jul 24 2012
a(n) = Sum_{k = 1..n} gcd(k, n)^2 * cos(2*Pi*k/n). - Enrique Pérez Herrero, Jan 18 2013
a(1) + a(2) + ... + a(n) ~ 1/(3*zeta(3))*n^3 + O(n^2). Lambert series Sum_{n >= 1} a(n)*x^n/(1 - x^n) = x*(1 + x)/(1 - x)^3. - Peter Bala, Dec 23 2013
n * a(n) = A000056(n). - Michael Somos, Mar 20 2004
a(n) = 24 * A115000(n) unless n < 5. - Michael Somos, Aug 12 2008
a(n) = A001065(n) - A134675(n). - Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
a(n) = Sum_{k=1..n} gcd(n, k) * phi(gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 15 2018
G.f.: Sum_{k>=1} mu(k)*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^2 - 1)^2) = 1.81078147612156295224312590448625180897250361794500723589001447178002894356... - Vaclav Kotesovec, Sep 19 2020
Limit_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k^2 = 1/zeta(3) (A088453). - Amiram Eldar, Oct 12 2020
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*mu(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} gcd(n,k)^2*mu(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} n*phi(gcd(n,k))/gcd(n,k).
a(n) = Sum_{k=1..n} phi(n*gcd(n,k))*mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} phi(n^2/gcd(n,k))*mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = Sum_{k = 1..n} phi(gcd(n, k)^2) = Sum_{d divides n} phi(d^2)*phi(n/d). - Peter Bala, Jan 17 2024
a(n) = Sum_{1 <= i, j <= n, lcm(i, j) = n} phi(i)*phi(j). See Tóth, p. 14. - Peter Bala, Jan 29 2024
Conjecture: a(n) = lim_{k->oo} (n^(2*(k + 1)))/A001157(n^k). - Velin Yanev, Dec 04 2024

Thanks to Michael Somos for catching an error in this sequence.

cp's OEIS Frontend

A001065 Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.

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A007434 Jordan function J_2(n) (a generalization of phi(n)).

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A134674 A134673 * A000012.

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