A083212 -id:A083212 - cp's OEIS frontend

A083206 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 17, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 14, 0, 0, 0, 1, 0, 13, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 2, 0, 1

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

a(n)=0 for deficient numbers n (A005100), but the converse is not true, as 18 is abundant (A005101) and a(18)=0, see A083211;
a(n)=1 for perfect numbers n (A000396), see A083209 for all numbers with a(n)=1;
records: A083213(k)=a(A083212(k)).
In order that a(n)>0, the sum of divisors of n must be even by definition: a(n) = half the number of partitions of A000203(n)/2 into divisors of n, see formula. [Reinhard Zumkeller, Jul 10 2010]

			a(24)=3: 1+2+3+4+8+12=6+24, 1+3+6+8+12=2+4+24, 4+6+8+12=1+2+3+24.

Antti Karttunen, Table of n, a(n) for n = 1..101632 (terms 1..800 from Reinhard Zumkeller, terms 801..10000 from T. D. Noe).
Reinhard Zumkeller, Illustration of initial terms

Cf. A083208 [= a(A083207(n))], A083211, A000005, A000203, A082729, A378446 (inverse Möbius transform), A378449.
Cf. A083207 (positions of terms > 0), A083210 (positions of 0's), A083209 (positions of 1's), A378652 (of 2's).
Cf. also A033630, A065205, A058377, A325806.

a[n_] := (s = DivisorSigma[1, n]; If[Mod[s, 2] == 1, 0, f[n, s/2, 2]]); 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[a, 105] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)

A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); }; \\ Antti Karttunen, Dec 02 2024, after Ilya Gutkovskiy

a(n) = if sigma(n) mod 2 = 1 then 0 else f(n,sigma(n)/2,2), where sigma=A000203 and 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, cf. A033630, also using f. [Reinhard Zumkeller, Jul 10 2010]

a(n) is half the coefficient of x^0 in Product_{d|n} (x^d + 1/x^d). - Ilya Gutkovskiy, Feb 04 2024

A083213 Record values in A083206.

Original entry on oeis.org

0, 1, 3, 5, 17, 140, 375, 1090, 11753, 373581, 1278347, 13637279, 163618153, 27924949575, 57188675627613, 610029725544935, 117122198574078151, 19988890974450988558, 239866235837773427391, 12281151167363439030630, 714539773759750811945040, 6986623779812404246388298, 1609715048049410062316011928, 5993993484832111940013400809611

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

A083206(k)A083212(n).

Antti Karttunen, Table of n, a(n) for n = 1..30

Cf. A083206, A083212.

a(n) = A083206(A083212(n)).

Extended by T. D. Noe, Jul 09 2010

More terms from Antti Karttunen, Dec 04 2024

A335144 Nonunitary Zumkeller numbers (A335142) whose set of nonunitary divisors can be partitioned into two disjoint sets of equal sum in a record number of ways.

Original entry on oeis.org

24, 96, 180, 216, 240, 360, 480, 720, 1080, 1440, 2160, 2880, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 45360, 50400, 60480, 75600, 100800, 110880, 151200, 221760, 277200, 302400, 332640, 453600, 498960, 554400, 665280, 831600, 1108800, 1330560

Offset: 1

Amiram Eldar, May 25 2020

nonn

The corresponding record values are 1, 3, 7, 13, 17, 102, 140, ... (see the link for more values).

			24 is the first term since it is the least nonunitary Zumkeller number, and its nonunitary divisors, {2, 4, 6, 12}, can be partitioned in a single way: 2 + 4 + 6 = 12. The next nonunitary Zumkeller number with more than one partition is 96, whose nonunitary divisors, {2, 4, 6, 8, 12, 16, 24, 48}, can be partitioned in 3 ways: 2 + 4 + 6 + 8 + 16 + 24 = 12 + 48, 2 + 6 + 12 + 16 + 24 = 4 + 8 + 48, and 8 + 12 + 16 + 24 = 2 + 4 + 6 + 48.

Amiram Eldar, Table of n, a(n), number of ways for n = 1..38

The nonunitary version of A083212.
Subsequence of A335142.
Cf. A335143.

nuz[n_] := Module[{d = Select[Divisors[n], GCD[#, n/#] > 1 &], sum, x}, sum = Plus @@ d; If[sum < 1 || OddQ[sum], 0, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]]/2]]; nuzm = 0; s = {}; Do[nuz1 = nuz[n]; If[nuz1 > nuzm, nuzm = nuz1; AppendTo[s, n]], {n, 1, 8000}]; s

A335220 Exponential Zumkeller numbers (A335218) whose set of exponential divisors can be partitioned into two disjoint sets of equal sum in a record number of ways.

Original entry on oeis.org

36, 900, 3600, 22500, 44100, 176400, 705600, 1587600, 4410000, 5336100, 21344400

Offset: 1

Amiram Eldar, May 27 2020

The corresponding record values are 1, 3, 4, 6, 83, 2920, 81080, 254566, 344022, 487267, 4580715031, ...

			36 is the first term since it is the least exponential Zumkeller number, and its exponential divisors, {6, 12, 18, 36}, can be partitioned in a single way: 6 + 12 + 18 = 36. The next exponential Zumkeller number with more than one partition is 900, whose nonunitary divisors, {30, 60, 90, 150, 180, 300, 450, 900}, can be partitioned in 3 ways: 30 + 60 + 90 + 150 + 300 + 450 = 180 + 900, 60 + 90 + 180 + 300 + 450 = 30 + 150 + 900, and 150 + 180 + 300 + 450 = 30 + 60 + 90 + 900.

The exponential version of A083212.
Subsequence of A335218.
Cf. A335219.

dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; eDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expDivQ[n, #] &]]; nways[n_] := Module[{d = eDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], 0, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]]/2]]; nwaysm = 0; s = {}; Do[nways1 = nways[n]; If[nways1 > nwaysm, nwaysm = nways1; AppendTo[s, n]], {n, 1, 23000}]; s

A348714 Numbers whose divisors can be partitioned into two disjoint sets with equal arithmetic mean in a record number of ways.

Original entry on oeis.org

1, 6, 24, 30, 60, 120, 168, 180, 240, 360, 420, 720, 840, 1260

Offset: 1

Amiram Eldar, Oct 31 2021

The corresponding record values are 0, 1, 2, 3, 19, 72, 99, 136, 248, 3094, 10452, 78057, 1323260, 4686578, ...

			6 is the smallest number whose set of divisors can be partitioned into two disjoint sets with equal arithmetic mean: {3} and {1, 2, 6}.
24 is the smallest number whose set of divisors can be partitioned into two disjoint sets with equal arithmetic mean in two ways: ({3, 12}, {1, 2, 4, 6, 8, 24}) and ({1, 2, 3, 24}, {4, 6, 8, 12}).

Cf. A027750, A083212, A348713.

c[n_] := Count[Subsets[(d = Divisors[n])], _?(Mean[#] == Mean[Complement[d, #]] &)]/2; cm = -1; s = {}; Do[If[(c1 = c[n]) > cm, cm = c1; AppendTo[s, n]], {n, 1, 250}]; s

cp's OEIS Frontend

A083206 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum.

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A083213 Record values in A083206.

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A335144 Nonunitary Zumkeller numbers (A335142) whose set of nonunitary divisors can be partitioned into two disjoint sets of equal sum in a record number of ways.

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A335220 Exponential Zumkeller numbers (A335218) whose set of exponential divisors can be partitioned into two disjoint sets of equal sum in a record number of ways.

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A348714 Numbers whose divisors can be partitioned into two disjoint sets with equal arithmetic mean in a record number of ways.

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