A027750 -id:A027750 - cp's OEIS frontend

A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.

6, 12, 20, 24, 28, 30, 40, 42, 48, 54, 56, 60, 66, 70, 78, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 150, 156, 160, 168, 174, 176, 180, 186, 192, 198, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270, 272

Offset: 1

Reinhard Zumkeller, Apr 22 2003

The 229026 Zumkeller numbers less than 10^6 have a maximum difference of 12. This leads to the conjecture that any 12 consecutive numbers include at least one Zumkeller number. There are 1989 odd Zumkeller numbers less than 10^6; they are exactly the odd abundant numbers that have even abundance, A174865. - T. D. Noe, Mar 31 2010
For k >= 0, numbers of the form 18k + 6 and 18k + 12 are terms (see Remark 2.3. in Somu et al., 2023). Corollary: The maximum difference between any two consecutive terms is at most 12. - Ivan N. Ianakiev, Jan 02 2024
All 205283 odd abundant numbers less than 10^8 that have even abundance (see A174865) are Zumkeller numbers. - T. D. Noe, Nov 14 2010
Except for 1 and 2, all primorials (A002110) are Zumkeller numbers (follows from Fact 6 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 23 2016
Supersequence of A111592 (follows from Fact 3 in the Rao/Peng paper). - Ivan N. Ianakiev, Mar 20 2017
Conjecture: Any 4 consecutive terms include at least one number k such that sigma(k)/2 is also a Zumkeller number (verified for the first 10^5 Zumkeller numbers). - Ivan N. Ianakiev, Apr 03 2017
LeVan studied these numbers using the equivalent definition of numbers n such that n = Sum_{d|n, dA180332) "minimal integer-perfect numbers". - Amiram Eldar, Dec 20 2018
The numbers 3 * 2^k for k > 0 are all Zumkeller numbers: half of one such partition is {3*2^k, 3*2^(k-2), ...}, replacing 3 with 2 if it appears. With this and the lemma that the product of a Zumkeller number and a number coprime to it is again a Zumkeller number (see A179527), we have that all numbers divisible by 6 but not 9 (or numbers congruent to 6 or 12 modulo 18) are Zumkeller numbers, proving that the difference between consecutive Zumkeller numbers is at most 12. - Charlie Neder, Jan 15 2019
Improvements on the previous comment: 1) For every integer q > 0, every odd integer r > 0 and every integer s > 0 relatively prime to 6, the integer 2^q*3^r*s is a Zumkeller number, and therefore 2) there exist Zumkeller numbers divisible by 9 (such as 54, 90, 108, 126, etc.). - Ivan N. Ianakiev, Jan 16 2020
Conjecture: If d > 1, d|k and tau(d)*sigma(d) = k, then k is a Zumkeller number (cf. A331668). - Ivan N. Ianakiev, Apr 24 2020
This sequence contains A378541, the intersection of the practical numbers (A005153) with numbers with even sum of divisors (A028983). - David A. Corneth, Nov 03 2024
Sequence gives the positions of even terms in A119347, and correspondingly, of odd terms in A308605. - Antti Karttunen, Nov 29 2024
If s = sigma(m) is odd and p > s then m*p is not in the sequence. - David A. Corneth, Dec 07 2024

			Given n = 48, we can partition the divisors thus: 1 + 3 + 4 + 6 + 8 + 16 + 24 = 2 + 12 + 48, therefore 48 is a term (A083206(48) = 5).
From _David A. Corneth_, Dec 04 2024: (Start)
30 is in the sequence. sigma(30) = 72. So we look for distinct divisors of 30 that sum to 72/2 = 36. That set or its complement contains 30. The other divisors in that set containing 30 sum to 36 - 30 = 6. So we look for some distinct proper divisors of 30 that sum to 6. That is from the divisors of {1, 2, 3, 5, 6, 10, 15}. It turns out that both 1+2+3 and 6 satisfy this condition. So 36 is in the sequence.
25 is not in the sequence as sigma(25) = 31 which is odd so the sum of two equal integers cannot be the sum of divisors of 25.
33 is not in the sequence as sigma(33) = 48 < 2*33. So is impossible to have a partition of the set of divisors into two disjoint set the sum of each of them sums to 48/2 = 24 as one of them contains 33 > 24 and any other divisors are nonnegative. (End)

Marijo O. LeVan, Integer-perfect numbers, Journal of Natural Sciences and Mathematics, Vol. 27, No. 2 (1987), pp. 33-50.
Marijo O. LeVan, On the order of nu(n), Journal of Natural Sciences and Mathematics, Vol. 28, No. 1 (1988), pp. 165-173.
J. Sandor and B. Crstici, Handbook of Number Theory, II, Springer Verlag, 2004, chapter 1.10, pp. 53-54.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Basher, k-Zumkeller labeling of super subdivision of some graphs, J. Egyptian Math. Soc. (2021) Vol. 29, No. 12.
Hussein Behzadipour, Two-layered numbers, arXiv:1812.07233 [math.NT], 2018.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, arXiv:0912.0052 [math.NT], 2009.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
David A. Corneth, PARI programs (Use function "is" for A179527)
Bhabesh Das, On unitary Zumkeller numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 2, 436-442.
Farid Jokar, On the difference between Zumkeller numbers, arXiv:1902.02168 [math.NT], 2019.
Farid Jokar, On k-layered numbers and some labeling related to k-layered numbers, arXiv:2003.11309 [math.NT], 2020.
Farid Jokar, On k-layered numbers, arXiv:2207.09053 [math.NT], 2022.
Peter Luschny, Zumkeller Numbers.
Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, arXiv:2008.11096 [math.NT], 2020.
Pankaj Jyoti Mahanta, Manjil P. Saikia, and Daniel Yaqubi, Some properties of Zumkeller numbers and k-layered numbers, Journal of Number Theory (2020).
Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, Some Results on Zumkeller Numbers, arXiv:2310.14149 [math.NT], 2023.
Reinhard Zumkeller, Illustration of initial terms
Index entries for sequences where any odd perfect numbers must occur

Positions of nonzero terms in A083206, positions of 0's in A103977 and in A378600.
Positions of even terms in A119347, of odd terms in A308605.
Complement of A083210.
Subsequence of A023196 and of A028983.
Union of A353061 and A378541.
Subsequences: A000396, A083209, A118372, A180332, A293453, A334410, A378652, A378599.
Conjectured subsequences: A007691, A331668 (after their initial 1's), A351548 (apart from 0-terms).
Cf. A174865 (Odd abundant numbers whose abundance is even).
Cf. A204830, A204831 (equal sums of 3 or 4 disjoint subsets).
Cf. A000203, A005101, A005153 (practical numbers), A005835, A027750, A048055, A083206, A083208, A083211, A171641, A175592, A179527 (characteristic function), A221054.

a083207 n = a083207_list !! (n-1)
a083207_list = filter (z 0 0 . a027750_row) $ [1..] where
   z u v []     = u == v
   z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
-- Reinhard Zumkeller, Apr 18 2013

with(numtheory): with(combstruct):
is_A083207 := proc(n) local S, R, Found, Comb, a, s; s := sigma(n);
if not(modp(s, 2) = 0 and n * 2 <= s) then return false fi;
S := s / 2 - n; R := select(m -> m <= S, divisors(n)); Found := false;
Comb := iterstructs(Combination(R)):
while not finished(Comb) and not Found do
   Found := add(a, a = nextstruct(Comb)) = S
od; Found end:
A083207_list := upto -> select(is_A083207, [$1..upto]):
A083207_list(272); # Peter Luschny, Dec 14 2009, updated Aug 15 2014

ZumkellerQ[n_] := Module[{d=Divisors[n], t, ds, x}, ds = Plus@@d; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; Select[Range[1000], ZumkellerQ] (* T. D. Noe, Mar 31 2010 *)
znQ[n_]:=Length[Select[{#,Complement[Divisors[n],#]}&/@Most[Rest[ Subsets[ Divisors[ n]]]],Total[#[[1]]]==Total[#[[2]]]&]]>0; Select[Range[300],znQ] (* Harvey P. Dale, Dec 26 2022 *)

part(n,v)=if(n<1, return(n==0)); forstep(i=#v,2,-1,if(part(n-v[i],v[1..i-1]), return(1))); n==v[1]
is(n)=my(d=divisors(n),s=sum(i=1,#d,d[i])); s%2==0 && part(s/2-n,d[1..#d-1]) \\ Charles R Greathouse IV, Mar 09 2014

PARI
```
\\ See Corneth link
```

from sympy import divisors
from sympy.combinatorics.subsets import Subset
for n in range(1,10**3):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and max(d) <= s/2:
        for x in range(1,2**len(d)):
            if sum(Subset.unrank_binary(x,d).subset) == s/2:
                print(n,end=', ')
                break
# Chai Wah Wu, Aug 13 2014

from sympy import divisors
import numpy as np
A083207 = []
for n in range(2,10**3):
    d = divisors(n)
    s = sum(d)
    if not s % 2 and 2*n <= s:
        d.remove(n)
        s2, ld = int(s/2-n), len(d)
        z = np.zeros((ld+1,s2+1),dtype=int)
        for i in range(1,ld+1):
            y = min(d[i-1],s2+1)
            z[i,range(y)] = z[i-1,range(y)]
            z[i,range(y,s2+1)] = np.maximum(z[i-1,range(y,s2+1)],z[i-1,range(0,s2+1-y)]+y)
            if z[i,s2] == s2:
                A083207.append(n)
                break
# Chai Wah Wu, Aug 19 2014

def is_Zumkeller(n):
    s = sigma(n)
    if not (2.divides(s) and n*2 <= s): return False
    S = s // 2 - n
    R = (m for m in divisors(n) if m <= S)
    return any(sum(c) == S for c in Combinations(R))
A083207_list = lambda lim: [n for n in (1..lim) if is_Zumkeller(n)]
print(A083207_list(272)) # Peter Luschny, Sep 03 2018

A083206(a(n)) > 0.
A083208(n) = A083206(a(n)).
A179529(a(n)) = 1. - Reinhard Zumkeller, Jul 19 2010

Name improved by T. D. Noe, Mar 31 2010

Name "Zumkeller numbers" added by N. J. A. Sloane, Jul 08 2010

A046951 a(n) is the number of squares dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Rediscovered by the HR automatic theory formation program.
a(n) depends only on prime signature of n (cf. A025487, A046523). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
First differences of A013936. Average value tends towards Pi^2/6 = 1.644934... (A013661, A013679). - Henry Bottomley, Aug 16 2001
We have a(n) = A159631(n) for all n < 125, but a(125) = 2 < 3 = A159631(125). - Steven Finch, Apr 22 2009
Number of 2-generated Abelian groups of order n, if n > 1. - Álvar Ibeas, Dec 22 2014 [In other words, number of order-n abelian groups with rank <= 2. Proof: let b(n) be such number. A finite abelian group is the inner direct product of all Sylow-p subgroups, so {b(n)} is multiplicative. Obviously b(p^e) = floor(e/2)+1 (corresponding to the groups C_(p^r) X C_(p^(e-r)) for 0 <= r <= floor(e/2)), hence b(n) = a(n) for all n. - Jianing Song, Nov 05 2022]
Number of ways of writing n = r*s such that r|s. - Eric M. Schmidt, Jan 08 2015
The number of divisors of the square root of the largest square dividing n. - Amiram Eldar, Jul 07 2020
The number of unordered factorizations of n into cubefree powers of primes (1, primes and squares of primes, A166684). - Amiram Eldar, Jun 12 2025

			a(16) = 3 because the squares 1, 4, and 16 divide 16.
G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Antonio Amariti, Claudius Klare, Domenico Orlando and Susanne Reffert, The M-theory origin of global properties of gauge theories, Nuclear Physics B, Vol. 901 (2015), pp. 318-337, arXiv preprint, arXiv:1507.04743 [hep-th], 2015 (see (A.13)).
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics
Ian G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. See delta(n).
Andrew V. Lelechenko, Average number of squares dividing mn, arXiv preprint arXiv:1407.1222 [math.NT], 2014.
Werner Georg Nowak and László Tóth, On the average number of subgroups of the group Z_m X Z_n, International Journal of Number Theory, Vol. 10, No. 2 (2014), pp. 363-374, arXiv preprint, arXiv:1307.1414 [math.NT], 2013.
N. J. A. Sloane, Transforms.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000188, A004101, A005117 (positions of 1's), A008619, A008833, A013936 (partial sums), A038538, A046952, A052304, A056595, A159631, A007814, A010052, A027750, A239930, A007862, A046523, A064989, A065704, A130279, A156552, A166684, A278161, A307445 (Dirichlet inverse).
One more than A071325.
Differs from A096309 for the first time at n=32, where a(32) = 3, while A096309(32) = 2 (and also A185102(32) = 2).
Sum of the k-th powers of the square divisors of n for k=0..10: this sequence (k=0), A035316 (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: this sequence (k=0), A340774 (k=1), A351600 (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).
Cf. A082293 (a(n)==2), A082294 (a(n)==3).

a046951 = sum . map a010052 . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[#[d: d in Divisors(n)|IsSquare(d)]:n in [1..120]]; // Marius A. Burtea, Jan 21 2020

A046951 := proc(n)
    local a,s;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*(1+floor(op(2,p)/2)) ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 17 2012
# Alternatively:
isbidivisible := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d:
a := n -> nops(select(k -> isbidivisible(n, k), [seq(1..n)])): # Peter Luschny, Jun 13 2025

a[n_] := Length[ Select[ Divisors[n], IntegerQ[Sqrt[#]]& ] ]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 26 2012 *)
Table[Length[Intersection[Divisors[n], Range[10]^2]], {n, 100}] (* Alonso del Arte, Dec 10 2012 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[ DivisorSigma[ 0, d], 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (Quotient[ #[[2]], 2] + 1 & /@ FactorInteger @ n)]; (* Michael Somos, Jun 13 2014 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^k^2 / (1 - x^k^2), {k, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Jun 13 2014 *)
f[p_, e_] := 1 + Floor[e/2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n)=my(f=factor(n));for(i=1,#f[,1],f[i,2]\=2);numdiv(factorback(f)) \\ Charles R Greathouse IV, Dec 11 2012

a(n) = direuler(p=2, n, 1/((1-X^2)*(1-X)))[n]; \\ Michel Marcus, Mar 08 2015

a(n)=factorback(apply(e->e\2+1, factor(n)[,2])) \\ Charles R Greathouse IV, Sep 17 2015

from math import prod
from sympy import factorint
def A046951(n): return prod((e>>1)+1 for e in factorint(n).values()) # Chai Wah Wu, Aug 04 2024

def is_bidivisible(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == d
def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_bidivisible(n, k)]
print([len(aList(n)) for n in range(1, 126)])  # Peter Luschny, Jun 13 2025

(definec (A046951 n) (if (= 1 n) 1 (* (A008619 (A007814 n)) (A046951 (A064989 n)))))
(define (A008619 n) (+ 1 (/ (- n (modulo n 2)) 2)))
;; Antti Karttunen, Nov 14 2016

a(p^k) = A008619(k) = [k/2] + 1. a(A002110(n)) = 1 for all n. (This is true for any squarefree number, A005117). - Original notes clarified by Antti Karttunen, Nov 14 2016
a(n) = |{(i, j) : i*j = n AND i|j}| = |{(i, j) : i*j^2 = n}|. Also tau(A000188(n)), where tau = A000005.
Multiplicative with p^e --> floor(e/2) + 1, p prime. - Reinhard Zumkeller, May 20 2007
a(A130279(n)) = n and a(m) <> n for m < A130279(n); A008966(n)=0^(a(n) - 1). - Reinhard Zumkeller, May 20 2007
Inverse Moebius transform of characteristic function of squares (A010052). Dirichlet g.f.: zeta(s)*zeta(2s).
G.f.: Sum_{k > 0} x^(k^2)/(1 - x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n) = Sum_{k=1..A000005(n)} A010052(A027750(n,k)). - Reinhard Zumkeller, Dec 16 2013
a(n) = Sum_{k = 1..n} ( floor(n/k^2) - floor((n-1)/k^2) ). - Peter Bala, Feb 17 2014
From Antti Karttunen, Nov 14 2016: (Start)
a(1) = 1; for n > 1, a(n) = A008619(A007814(n)) * a(A064989(n)).
a(n) = A278161(A156552(n)). (End)
G.f.: Sum_{k>0}(theta(q^k)-1)/2, where theta(q)=1+2q+2q^4+2q^9+2q^16+... - Mamuka Jibladze, Dec 04 2016
From  Antti Karttunen, Nov 12 2017: (Start)
a(n) = A000005(n) - A056595(n).
a(n) = 1 + A071325(n).
a(n) = 1 + A001222(A293515(n)). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A000005(d) * A008836(n/d). - Torlach Rush, Jan 21 2020
a(n) = A000005(sqrt(A008833(n))). - Amiram Eldar, Jul 07 2020
a(n) = Sum_{d divides n} mu(core(d)^2), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

Data section filled up to 125 terms and wrong claim deleted from Crossrefs section by Antti Karttunen, Nov 14 2016

A003325 Numbers that are the sum of 2 positive cubes.

Original entry on oeis.org

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343

Offset: 1

N. J. A. Sloane

It is conjectured that this sequence and A052276 have infinitely many numbers in common, although only one example (128) is known. [Any further examples are greater than 5 million. - Charles R Greathouse IV, Apr 12 2020] [Any further example is greater than 10^12. - M. F. Hasler, Jan 10 2021]
A113958 is a subsequence; if m is a term then m+k^3 is a term of A003072 for all k > 0. - Reinhard Zumkeller, Jun 03 2006
From James R. Buddenhagen, Oct 16 2008: (Start)
(i) N and N+1 are both the sum of two positive cubes if N=2*(2*n^2 + 4*n + 1)*(4*n^4 + 16*n^3 + 23*n^2 + 14*n + 4), n=1,2,....
(ii) For n >= 2, let N = 16*n^6 - 12*n^4 + 6*n^2 - 2, so N+1 = 16*n^6 - 12*n^4 + 6*n^2 - 1.
Then the identities 16*n^6 - 12*n^4 + 6*n^2 - 2 = (2*n^2 - n - 1)^3 + (2*n^2 + n - 1)^3 16*n^6 - 12*n^4 + 6*n^2 - 1 = (2*n^2)^3 + (2*n^2 - 1)^3 show that N, N+1 are in the sequence. (End)
If n is a term then n*m^3 (m >= 2) is also a term, e.g., 2m^3, 9m^3, 28m^3, and 35m^3 are all terms of the sequence. "Primitive" terms (not of the form n*m^3 with n = some previous term of the sequence and m >= 2) are 2, 9, 28, 35, 65, 91, 126, etc. - Zak Seidov, Oct 12 2011
This is an infinite sequence in which the first term is prime but thereafter all terms are composite. - Ant King, May 09 2013
By Fermat's Last Theorem (the special case for exponent 3, proved by Euler, is sufficient), this sequence contains no cubes. - Charles R Greathouse IV, Apr 03 2021

C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 354.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
C. G. J. Jacobi, Gesammelte Werke.
Michael Penn, 1674 is not a perfect cube, 2020 video
N. J. A. Sloane, Table of n, a(n) for n = 1..59562
D. Tournes, A Glance on Indian Mathematician Srinivasa Ramanujan(1887-1920). [Text in French]
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Subsequence of A004999 and hence of A045980; supersequence of A202679.

Cf. A024670 (2 distinct cubes), A003072, A001235, A011541, A003826, A010057, A000578, A027750, A010052, A085323 (n such that a(n+1)=a(n)+1).

a003325 n = a003325_list !! (n-1)
a003325_list = filter c2 [1..] where
   c2 x = any (== 1) $ map (a010057 . fromInteger) $
                       takeWhile (> 0) $ map (x -) $ tail a000578_list
-- Reinhard Zumkeller, Mar 24 2012

nn = 2*20^3; Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]] (* T. D. Noe, Oct 12 2011 *)
With[{upto=2000},Select[Total/@Tuples[Range[Ceiling[Surd[upto,3]]]^3,2],#<=upto&]]//Union (* Harvey P. Dale, Jun 11 2016 *)

cubes=sum(n=1, 11, x^(n^3), O(x^1400)); v = select(x->x, Vec(cubes^2), 1); vector(#v, k, v[k]+1) \\ edited by Michel Marcus, May 08 2017

isA003325(n) = for(k=1,sqrtnint(n\2,3), ispower(n-k^3,3) && return(1)) \\ M. F. Hasler, Oct 17 2008, improved upon suggestion of Altug Alkan and Michel Marcus, Feb 16 2016

T=thueinit('z^3+1); is(n)=#select(v->min(v[1],v[2])>0, thue(T,n))>0 \\ Charles R Greathouse IV, Nov 29 2014

list(lim)=my(v=List()); lim\=1; for(x=1,sqrtnint(lim-1,3), my(x3=x^3); for(y=1,min(sqrtnint(lim-x3,3),x), listput(v, x3+y^3))); Set(v) \\ Charles R Greathouse IV, Jan 11 2022

from sympy import integer_nthroot
def aupto(lim):
  cubes = [i*i*i for i in range(1, integer_nthroot(lim-1, 3)[0] + 1)]
  sum_cubes = sorted([a+b for i, a in enumerate(cubes) for b in cubes[i:]])
  return [s for s in sum_cubes if s <= lim]
print(aupto(1343)) # Michael S. Branicky, Feb 09 2021

Error in formula line corrected by Zak Seidov, Jul 23 2009

A017666 Denominator of sum of reciprocals of divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 6, 19, 10, 21, 11, 23, 2, 25, 13, 27, 1, 29, 5, 31, 32, 11, 17, 35, 36, 37, 19, 39, 4, 41, 7, 43, 11, 15, 23, 47, 12, 49, 50, 17, 26, 53, 9, 55, 7, 57, 29, 59, 5, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35, 71, 24, 73, 37, 75, 19

Offset: 1

N. J. A. Sloane

Sum_{ d divides 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 (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
Denominators of coefficients in expansion of Sum_{n >= 1} x^n/(n*(1-x^n)) = Sum_{n >= 1} log(1/(1-x^n)).
Also n/gcd(n, sigma(n)) = n/A009194(n); also n/lcm(all common divisors of n and sigma(n)). Equals 1 if 6,28,120,496,672,8128,..., i.e., if n is from A007691. - Labos Elemer, Aug 14 2002
a(A007691(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Denominator of sigma(n)/n = A000203(n)/n. a(n) = 1 for numbers n in A007691 (multiply-perfect numbers), a(n) = 2 for numbers n in A159907 (numbers n with half-integral abundancy index), a(n) = 3 for numbers n in A245775, a(n) = n for numbers n in A014567 (numbers n such that n and sigma(n) are relatively prime). See A162657 (n) - the smallest number k such that a(k) = n. - Jaroslav Krizek, Sep 23 2014
For all n, a(n) <= n, and thus records are obtained for terms of A014567. - Michel Marcus, Sep 25 2014
Conjecture: If a(n) is in A005153, then n is in A005153. In particular, if n has dyadic rational abundancy index, i.e., a(n) is in A000079 (such as A007691 and A159907), then n is in A005153. Since every term of A005153 greater than 1 is even, any odd n such that a(n) in A005153 must be in A007691. It is natural to ask if there exists a generalization of the indicator function for A005153, call it m(n), such that m(n) = 1 for n in A005153, 0 < m(n) < 1 otherwise, and m(a(n)) <= m(n) for all n. See also A050972. - Jaycob Coleman, Sep 27 2014

			1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, 12/7, 8/5, 31/16, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 4th formula.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Abundancy

Cf. A017665, A027750.

import Data.Ratio ((%), denominator)
a017666 = denominator . sum . map (1 %) . a027750_row
-- Reinhard Zumkeller, Apr 06 2012

[Denominator(DivisorSigma(1,n)/n): n in [1..50]]; // G. C. Greubel, Nov 08 2018

with(numtheory): seq(denom(sigma(n)/n), n=1..76) ; # Zerinvary Lajos, Jun 04 2008

Table[Denominator[DivisorSigma[-1, n]], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2011 *)
Table[Denominator[DivisorSigma[1, n]/n], {n, 1, 50}] (* G. C. Greubel, Nov 08 2018 *)

a(n) = denominator(sigma(n)/n); \\ Michel Marcus, Sep 23 2014

from math import gcd
from sympy import divisor_sigma
def A017666(n): return n//gcd(divisor_sigma(n),n) # Chai Wah Wu, Mar 21 2023

More terms from Labos Elemer, Aug 14 2002

A001599 Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.

Original entry on oeis.org

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720

Offset: 1

N. J. A. Sloane

Note that the harmonic mean of the divisors of k = k*tau(k)/sigma(k).
Equivalently, k*tau(k)/sigma(k) is an integer, where tau(k) (A000005) is the number of divisors of k and sigma(k) is the sum of the divisors of k (A000203).
Equivalently, the average of the divisors of k divides k.
Note that the average of the divisors of k is not necessarily an integer, so the above wording should be clarified as follows: k divided by the average is an integer. See A007340. - Thomas Ordowski, Oct 26 2014
Ore showed that every perfect number (A000396) is harmonic. The converse does not hold: 140 is harmonic but not perfect. Ore conjectured that 1 is the only odd harmonic number.
Other examples of power mean numbers k such that some power mean of the divisors of k is an integer are the RMS numbers A140480. - Ctibor O. Zizka, Sep 20 2008
Conjecture: Every harmonic number is practical (A005153). I've verified this refinement of Ore's conjecture for all terms less than 10^14. - Jaycob Coleman, Oct 12 2013
Conjecture: All terms > 1 are Zumkeller numbers (A083207). Verified for all n <= 50. - Ivan N. Ianakiev, Nov 22 2017
Verified for n <= 937. - David A. Corneth, Jun 07 2020
Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0. - Amiram Eldar, Jun 01 2020
Zachariou and Zachariou (1972) called these numbers "Ore numbers", after the Norwegian mathematician Øystein Ore (1899 - 1968), who was the first to study them. Ore (1948) and Garcia (1954) referred to them as "numbers with integral harmonic mean of divisors". The term "harmonic numbers" was used by Pomerance (1973). They are sometimes called "harmonic divisor numbers", or "Ore's harmonic numbers", to differentiate them from the partial sums of the harmonic series. - Amiram Eldar, Dec 04 2020
Conjecture: all terms > 1 have a Mersenne prime as a factor. - Ivan Borysiuk, Jan 28 2024

			k=140 has sigma_0(140)=12 divisors with sigma_1(140)=336. The average divisor is 336/12=28, an integer, and divides k: k=5*28, so 140 is in the sequence.
k=496 has sigma_0(496)=10, sigma_1(496)=992: the average divisor 99.2 is not an integer, but k/(sigma_1/sigma_0)=496/99.2=5 is an integer, so 496 is in the sequence.

G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.
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 147.

Robert G. Wilson v, Table of n, a(n) for n = 1..937 (terms n = 1..170 from T. D. Noe and Klaus Brockhaus)
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Ivan Borysiuk, Conjectured to be the 10000 smallest Ore numbers
Graeme L. Cohen, Email to N. J. A. Sloane, Apr. 1994.
Graeme L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Mathematics of Computation, Vol. 66, No. 218, (1997), pp. 883-891.
Graeme L. Cohen and Ronald M. Sorli, Harmonic seeds, Fibonacci Quart., Vol. 36, No. 5 (1998), pp. 386-390; errata, 39 (2001) 4.
Graeme L. Cohen and Ronald M. Sorli, Odd harmonic numbers exceed 10^24, Math. Comp., Vol. 79, No. 272 (2010), pp. 2451-2460.
Mariano Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly, Vol. 61, No. 2 (1954), pp. 89-96.
Takeshi Goto, All harmonic numbers less than 10^14.
Takeshi Goto, Table of a(n) for n = 1..937.
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput., Vol. 73, No. 245 (2004), pp. 475-491.
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371-374.
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.
Oystein Ore, On the averages of the divisors of a number. (annotated scanned copy)
Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709-A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A-648, entire volume.
Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Harmonic Divisor Number.
Wikipedia, Harmonic mean.
Wikipedia, Harmonic divisor number.
Andreas and Eleni Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.
Index entries for sequences where any odd perfect numbers must occur

See A003601 for analogs referring to arithmetic mean and A000290 for geometric mean of divisors.
See A001600 and A090240 for the integer values obtained.
sigma_0(n) (or tau(n)) is the number of divisors of n (A000005).
sigma_1(n) (or sigma(n)) is the sum of the divisors of n (A000203).
Cf. A007340, A090945, A035527, A007691, A074247, A053783. Not a subset of A003601.
Cf. A027750.

Concatenation([1],Filtered([2,4..2000000],n->IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018

import Data.Ratio (denominator)
import Data.List (genericLength)
a001599 n = a001599_list !! (n-1)
a001599_list = filter ((== 1) . denominator . hm) [1..] where
   hm x = genericLength ds * recip (sum $ map (recip . fromIntegral) ds)
          where ds = a027750_row x
-- Reinhard Zumkeller, Jun 04 2013, Jan 20 2012

q:= (p,k) -> p^k*(p-1)*(k+1)/(p^(k+1)-1):
filter:= proc(n) local t; mul(q(op(t)),t=ifactors(n)[2])::integer end proc:
select(filter, [$1..10^6]); # Robert Israel, Jan 14 2016

Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]
Select[Range[1600000],IntegerQ[HarmonicMean[Divisors[#]]]&] (* Harvey P. Dale, Oct 20 2012 *)

a(n)=if(n<0,0,n=a(n-1);until(0==(sigma(n,0)*n)%sigma(n,1),n++);n) /* Michael Somos, Feb 06 2004 */

from sympy import divisor_sigma as sigma
def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

from itertools import count, islice
from functools import reduce
from math import prod
from sympy import factorint
def A001599_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
            yield n
A001599_list = list(islice(A001599_gen(),20)) # Chai Wah Wu, Feb 14 2023

{ k : A106315(k) = 0 }. - R. J. Mathar, Jan 25 2017

More terms from Klaus Brockhaus, Sep 18 2001

A077610 Triangle in which n-th row lists unitary divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 3, 8, 24, 1, 25, 1, 2, 13, 26, 1, 27, 1, 4, 7, 28, 1, 29, 1, 2, 3, 5, 6, 10, 15, 30

Offset: 1

Eric W. Weisstein, Nov 11 2002

n-th row = n-th row of A165430 without repetitions. - Reinhard Zumkeller, Mar 04 2013
Denominators of sequence of all positive rational numbers ordered as follows:  let m = p(i(1))^e(i(1))*...*p(i(k))^e(i(k)) be the prime factorization of m.  Let S(m) be the vector of rationals p(i(k+1-j))^e(i(k+1-j))/p(i(j))^e(i(j)) for j = 1..k.  The sequence (a(n)) is the concatenation of vectors S(m) for m = 1, 2, ...; for numerators see A229994. - Clark Kimberling, Oct 31 2013
The concept of unitary divisors was introduced by the Indian mathematician Ramaswamy S. Vaidyanathaswamy (1894-1960) in 1931. He called them "block factors". The term "unitary divisor" was coined by Cohen (1960). - Amiram Eldar, Mar 09 2024

			1;
1, 2;
1, 3;
1, 4;
1, 5;
1, 2, 3, 6;
1, 7;
1, 8;
1, 9;
1, 2, 5, 10;
1, 11;

Reinhard Zumkeller, Rows n=1..1000 of triangle, flattened
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
R. Vaidyanathaswamy, The theory of multiplicative arithmetic functions, Transactions of the American Mathematical Society, Vol. 33, No. 2 (1931), pp. 579-662.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Index entries for sequences related to enumerating the rationals.

Cf. A037445, A027750, A034444 (row lengths), A034448 (row sums); A206778.

a077610 n k = a077610_row n !! k
a077610_row n = [d | d <- [1..n], let (n',m) = divMod n d,
                     m == 0, gcd d n' == 1]
a077610_tabf = map a077610_row [1..]
-- Reinhard Zumkeller, Feb 12 2012

with(numtheory);
# returns the number of unitary divisors of n and a list of them, from N. J. A. Sloane, May 01 2013
f:=proc(n)
local ct,i,t1,ans;
ct:=0; ans:=[];
t1:=divisors(n);
for i from 1 to nops(t1) do
d:=t1[i];
if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi;
od:
RETURN([ct,ans]);
end;
# Alternatively:
isUnitary := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = 1:
aList := n -> select(k -> isUnitary(n, k), [seq(1..n)]): # Peter Luschny, Jun 13 2025

row[n_] := Select[ Divisors[n], GCD[#, n/#] == 1 &]; Table[row[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Oct 22 2012 *)

row(n)=my(f=factor(n),k=#f~); Set(vector(2^k,i, prod(j=1,k, if(bittest(i,j-1),1,f[j,1]^f[j,2]))))
v=[];for(n=1,20,v=concat(v,row(n)));v \\ Charles R Greathouse IV, Sep 02 2015

row(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d);} \\ Michel Marcus, Oct 11 2015

from math import gcd
def is_unitary(n, d) -> bool: return gcd(n, d) == d and gcd(n//d, d) == 1
def aList(n) -> list[int]: return [k for k in range(1, n+1) if is_unitary(n, k)]
for n in range(1, 31): print(aList(n))  # Peter Luschny, Jun 13 2025

d is unitary divisor of n <=> gcd(n, d) = d and gcd(n/d, d) = 1. - Peter Luschny, Jun 13 2025

A018818 Number of partitions of n into divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 10, 5, 11, 2, 45, 2, 14, 14, 36, 2, 81, 2, 92, 18, 20, 2, 458, 7, 23, 23, 156, 2, 742, 2, 202, 26, 29, 26, 2234, 2, 32, 30, 1370, 2, 1654, 2, 337, 286, 38, 2, 9676, 9, 407, 38, 454, 2, 3132, 38, 3065, 42, 47, 2, 73155, 2, 50, 493, 1828, 44, 5257, 2, 740, 50, 5066

Offset: 1

David W. Wilson

From Reinhard Zumkeller, Dec 11 2009: (Start)
For odd primes p: a(p^2) = p + 2; for n > 1: a(A001248(n)) = A052147(n);
For odd primes p > 3, a(3*p) = 2*p + 4; for n > 2: a(A001748(n)) = A100484(n) + 4. (End)
From Matthew Crawford, Jan 19 2021: (Start)
For a prime p, a(p^3) = (p^3 + p^2 + 2*p + 4)/2;
For distinct primes p and q, a(p*q) = (p+1)*(q+1)/2 + 2. (End)

			The a(6) = 8 representations of 6 are 6 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Douglas Bowman et al., Problem 6640: Partitions of n into parts which are divisors of n, American Mathematical Monthly, 99(3) (1992), 276-277.
Hansraj Gupta, Partitions of n into divisors of m, Indian J. Pure Appl. Math., 6(11) (1975), 1276-1286.
Hansraj Gupta, Partitions of n into divisors of m, Indian J. Pure Appl. Math., 6(11) (1975), 1276-1286.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Noah Lebowitz-Lockard and Joseph Vandehey, On the number of partitions of a number into distinct divisors, arXiv:2402.08119 [math.NT], 2024. See p. 1.
Rémy Sigrist, Colored logarithmic scatterplot of the first 10000 terms (where the color is function of A000005(n)).
Index entries for sequences related to partitions.

Cf. A002577, A027750, A033630, A072721, A161148 (partitions in squared divisors), A171565, A210442, A211110, A225244, A306387, A379957.

Cf. A000005, A001248, A001748, A052147, A100484.

a018818 n = p (init $ a027750_row n) n + 1 where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 02 2012

[#RestrictedPartitions(n,{d:d in Divisors(n)}): n in [1..100]]; // Marius A. Burtea, Jan 02 2019

A018818 := proc(n)
    local a,p,w,el ;
    a := 0 ;
    for p in combinat[partition](n) do
        w := true ;
        for el in p do
            if modp(n,el) <> 0 then
                w := false;
                break;
            end if;
        end do:
        if w then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 30 2017

Table[d = Divisors[n]; Coefficient[Series[1/Product[1 - x^d[[i]], {i, Length[d]}], {x, 0, n}], x, n], {n, 100}] (* T. D. Noe, Jul 28 2011 *)

a(n)=numbpartUsing(n, divisors(n));
numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1,mx, numbpartUsing(n-v[i],v,i)) \\ inefficient; Charles R Greathouse IV, Jun 21 2017

A018818(n) = { my(p = Ser(1, 'x, 1+n)); fordiv(n, d, p /= (1 - 'x^d)); polcoef(p, n); }; \\ Antti Karttunen, Jan 23 2025, after Vladeta Jovovic

Coefficient of x^n in the expansion of 1/Product_{d|n} (1-x^d). - Vladeta Jovovic, Sep 28 2002
a(n) = 2 iff n is prime. - Juhani Heino, Aug 27 2009
a(n) = f(n,n,1), where f(n,m,k) = f(n,m,k+1) + f(n,m-k,k)*0^(n mod k) if k <= m, otherwise 0^m. - Reinhard Zumkeller, Dec 11 2009
Paul Erdős, Andrew M. Odlyzko, and the Editors of the AMM give bounds; see Bowman et al. - Charles R Greathouse IV, Dec 04 2012

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.

A048050 Chowla's function: sum of divisors of n except for 1 and n.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 6, 3, 7, 0, 15, 0, 9, 8, 14, 0, 20, 0, 21, 10, 13, 0, 35, 5, 15, 12, 27, 0, 41, 0, 30, 14, 19, 12, 54, 0, 21, 16, 49, 0, 53, 0, 39, 32, 25, 0, 75, 7, 42, 20, 45, 0, 65, 16, 63, 22, 31, 0, 107, 0, 33, 40, 62, 18, 77, 0, 57, 26, 73, 0, 122, 0, 39, 48, 63, 18, 89

Offset: 1

N. J. A. Sloane

a(n) = 0 if and only if n is a noncomposite number (cf. A008578). - Omar E. Pol, Jul 31 2012
If n is semiprime, a(n) = A008472(n). - Wesley Ivan Hurt, Aug 22 2013
If n = p*q where p and q are distinct primes then a(n) = p+q.
If k,m > 1 are coprime, then a(k*m) = a(k)*a(m) + (m+1)*a(k) + (k+1)*a(m) + k + m. - Robert Israel, Apr 28 2015
a(n) is also the total number of parts in the partitions of n into equal parts that contain neither 1 nor n as a part (see example). More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that contain neither k nor k*n as a part. - Omar E. Pol, Nov 24 2019
Named after the Indian-American mathematician Sarvadaman D. S. Chowla (1907-1995). - Amiram Eldar, Mar 09 2024

			For n = 20 the divisors of 20 are 1,2,4,5,10,20, so a(20) = 2+4+5+10 = 21.
On the other hand, the partitions of 20 into equal parts that contain neither 1 nor 20 as a part are [10,10], [5,5,5,5], [4,4,4,4,4], [2,2,2,2,2,2,2,2,2,2]. There are 21 parts, so a(20) = 21. - _Omar E. Pol_, Nov 24 2019

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., 25 (1971), 923-925.
Abdur Rahman Nasir, On a certain arithmetic function, Bull. Calcutta Math. Soc., Vol. 38 (1946), p. 140.
Omar E. Pol, Illustration of initial terms obtained geometrically.

Cf. A000203, A001065, A000593, A002954, A048995, A007956, A048671, A182936.

Cf. A057533, A005276, A027751, A237593, A245092, A244049 (partial sums).

a048050 1 = 0
a048050 n = (subtract 1) $ sum $ a027751_row n
-- Reinhard Zumkeller, Feb 09 2013

A048050:=func< n | n eq 1 or IsPrime(n) select 0 else &+[ a: a in Divisors(n) | a ne 1 and a ne n ] >; [ A048050(n): n in [1..100] ]; // Klaus Brockhaus, Mar 04 2011

A048050 := proc(n) if n > 1 then numtheory[sigma](n)-1-n ; else 0; end if; end proc:

f[n_]:=Plus@@Divisors[n]-n-1; Table[f[n],{n,100}] (*Vladimir Joseph Stephan Orlovsky, Sep 13 2009*)
Join[{0},DivisorSigma[1,#]-#-1&/@Range[2,80]] (* Harvey P. Dale, Feb 25 2015 *)

a(n)=if(n>1,sigma(n)-n-1,0) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import divisors
def a(n): return sum(divisors(n)[1:-1]) # Indranil Ghosh, Apr 26 2017

from sympy import divisor_sigma
def A048050(n): return 0 if n == 1 else divisor_sigma(n)-n-1 # Chai Wah Wu, Apr 18 2021

a(n) = A000203(n) - A065475(n).
a(n) = A001065(n) - 1, n > 1.
For n > 1: a(n) = Sum_{k=2..A000005(n)-1} A027750(n,k). - Reinhard Zumkeller, Feb 09 2013
a(n) = A000203(n) - n - 1, n > 1. - Wesley Ivan Hurt, Aug 22 2013
G.f.: Sum_{k>=2} k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Jan 22 2017

A033630 Number of partitions of n into distinct divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 35, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 2, 1, 7, 1, 1, 1, 26, 1, 1, 1, 2, 1, 24, 1, 1, 1, 1, 1, 22, 1, 1, 1, 3

Offset: 0

Marc LeBrun

nonn

			a(12) = 3 because we have the partitions [12], [6, 4, 2], and [6, 3, 2, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (1000 terms from T. D. Noe)
Noah Lebowitz-Lockard and Joseph Vandehey, On the number of partitions of a number into distinct divisors, arXiv:2402.08119 [math.NT], 2024. See p. 2.

Cf. A018818, A065205.
Cf. A083206. - Reinhard Zumkeller, Jul 19 2010
Cf. A000009, A005153.
Cf. A211111, A027750.
Cf. A225245.
Cf. A000396, A005100, A005835, A006037.

a033630 0 = 1
a033630 n = p (a027750_row n) n where
   p _  0 = 1
   p [] _ = 0
   p (d:ds) m = if d > m then 0 else p ds (m - d) + p ds m
-- Reinhard Zumkeller, Feb 23 2014, Apr 04 2012, Oct 27 2011

with(numtheory): a:=proc(n) local div, g, gser: div:=divisors(n): g:=product(1+x^div[j],j=1..tau(n)): gser:=series(g,x=0,105): coeff(gser,x^n): end: seq(a(n),n=1..100); # Emeric Deutsch, Mar 30 2006
# second Maple program:
with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n))[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i-1))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100); # Alois P. Heinz, Feb 05 2014

A033630 = Table[SeriesCoefficient[Series[Times@@((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n ], {n, 512}] (* Wouter Meeussen *)
A033630[n_] := f[n, n, 1]; f[n_, m_, k_] := f[n, m, k] = If[k <= m, f[n, m, k + 1] + f[n, m - k, k + 1] * Boole[Mod[n, k] == 0], Boole[m == 0]]; Array[A033630, 101, 0] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)

a(n) = A065205(n) + 1.
a(A005100(n)) = 1; a(A005835(n)) > 1. - Reinhard Zumkeller, Mar 02 2007
a(n) = f(n, n, 1) with f(n, m, k) = if k <= m then f(n, m, k + 1) + f(n, m - k, k + 1)*0^(n mod k) else 0^m. - Reinhard Zumkeller, Dec 11 2009
a(n) = [x^n] Product_{d|n} (1 + x^d). - Ilya Gutkovskiy, Jul 26 2017
a(n) = 1 if n is deficient (A005100) or weird (A006037). a(n) = 2 if n is perfect (A000396). - Alonso del Arte, Sep 24 2017

More terms from Reinhard Zumkeller, Apr 21 2003

cp's OEIS Frontend

A083207 Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.

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A046951 a(n) is the number of squares dividing n.

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A003325 Numbers that are the sum of 2 positive cubes.

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A017666 Denominator of sum of reciprocals of divisors of n.

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A001599 Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.

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