cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A335143 Nonunitary Zumkeller numbers (A335142) whose set of nonunitary divisors can be partitioned into two disjoint sets of equal sum in a single way.

Original entry on oeis.org

24, 48, 54, 80, 112, 150, 224, 280, 294, 352, 416, 630, 704, 726, 832, 1014, 1088, 1216, 1472, 1734, 1750, 1856, 1984, 2166, 2475, 2944, 3174, 3344, 3430, 3712, 3968, 4275, 4736, 5046, 5248, 5504, 5766, 6016, 6784, 7552, 7808, 8214, 8470, 10086, 11008, 11094
Offset: 1

Views

Author

Amiram Eldar, May 25 2020

Keywords

Examples

			24 is a term since there is only one partition of its set of nonunitary divisors, {2, 4, 6, 12}, into two disjoint sets of equal sum: {2, 4, 6} and {12}.

Crossrefs

The nonunitary version of A083209.
Subsequence of A335142.

Programs

  • Mathematica
    nuzQ[n_] := Module[{d = Select[Divisors[n], GCD[#, n/#] > 1 &], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 2]; Select[Range[12000], nuzQ]

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

Views

Author

Amiram Eldar, May 25 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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

A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 88, 90, 96, 102, 104, 114, 120, 138, 150, 160, 162, 168, 174, 186, 192, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390, 402
Offset: 1

Views

Author

Amiram Eldar, May 27 2020

Keywords

Examples

			6 is a term since its set of bi-unitary divisors, {1, 2, 3, 6}, can be partitioned into 2 disjoint sets, whose sum is equal: 1 + 2 + 3 = 6.

Links

Crossrefs

The bi-unitary version of A083207.
Subsequence of A292982.
Cf. A188999, A222266, A290466, A323342, A335142, A335197.

Programs

  • Mathematica
    uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[10^3], bzQ]

A339979 Coreful Zumkeller numbers: numbers whose set of coreful divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

36, 72, 144, 180, 200, 252, 288, 324, 360, 392, 396, 400, 468, 504, 576, 600, 612, 648, 684, 720, 784, 792, 800, 828, 900, 936, 1008, 1044, 1116, 1152, 1176, 1200, 1224, 1260, 1296, 1332, 1368, 1400, 1440, 1476, 1548, 1568, 1584, 1600, 1620, 1656, 1692, 1764
Offset: 1

Views

Author

Amiram Eldar, Dec 25 2020

Keywords

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
The coreful perfect numbers (A307958) are a subsequence.

Examples

			36 is a term since its set of coreful divisors, {6, 12, 18, 36}, can be partitioned into the two disjoint sets, {6, 12, 18} and {36}, whose sums are equal: 6 + 12 + 18 = 36.

Crossrefs

A307958 is a subsequence.
Subsequence of A308053.
Cf. A007947, A057723.
Similar sequences: A083207, A290466, A335197, A335142, A335215, A335218.

Programs

  • Mathematica
    corZumQ[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r * Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1800], corZumQ]
  • Python
    from itertools import count, islice
from sympy import primefactors, divisors
def A339979_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = primefactors(n)
        d = [x for x in divisors(n) if primefactors(x)==f]
        s = sum(d)
        if s&1^1 and n<<1<=s:
            d = d[:-1]
            s2, ld = (s>>1)-n, len(d)
            z = [[0 for  in range(s2+1)] for  in range(ld+1)]
            for i in range(1, ld+1):
                y = min(d[i-1], s2+1)
                z[i][:y] = z[i-1][:y]
                for j in range(y,s2+1):
                    z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
                if z[i][s2] == s2:
                    yield n
                    break
A339979_list = list(islice(A339979_gen(),20)) # Chai Wah Wu, Feb 14 2023

A348527 Noninfinitary Zumkeller numbers: numbers whose set of noninfinitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

48, 80, 96, 112, 150, 180, 240, 252, 294, 336, 360, 396, 432, 468, 480, 486, 504, 528, 560, 600, 612, 624, 630, 672, 684, 720, 726, 768, 792, 810, 816, 828, 864, 880, 912, 936, 960, 1008, 1014, 1040, 1044, 1050, 1056, 1104, 1116, 1120, 1134, 1176, 1200, 1232, 1248
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Comments

The smallest odd term is a(104) = 2475.

Examples

			48 is a term since its set of noninfinitary divisors, {2, 4, 6, 8, 12, 24}, can be partitioned into the two disjoint sets, {2, 6, 8, 12} and {4, 24}, whose sums are equal: 2 + 6 + 8 + 12 = 4 + 24 = 28.

Links

Crossrefs

Cf. A348271, A348274, A348341.
Similar sequences: A083207, A290466, A335215, A335142, A335197, A335218.

Programs

  • Mathematica
    nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; nizQ[n_] := Module[{d = nidiv[n], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1250], !IntegerQ@ Log2@ DivisorSigma[0, #] && nizQ[#] &]

A335145 Numbers that are both unitary and nonunitary Zumkeller numbers.

Original entry on oeis.org

150, 294, 630, 726, 750, 840, 1014, 1050, 1470, 1650, 1734, 1890, 1950, 2058, 2166, 2550, 2850, 2940, 2970, 3174, 3234, 3450, 3510, 3630, 3750, 3822, 4350, 4410, 4650, 4998, 5046, 5070, 5082, 5250, 5550, 5586, 5670, 5766, 6150, 6450, 6762, 6930, 7050, 7098, 7260
Offset: 1

Views

Author

Amiram Eldar, May 25 2020

Keywords

Examples

			150 is a term since its unitary divisors, {1, 2, 3, 6, 25, 50, 75, 150} can be partitioned in two disjoint sets of equal sum: 1 + 2 + 3 + 25 + 50 + 75 = 6 + 150, and so are its nonunitary divisors, {5, 10, 15, 30}: 5 + 10 + 15 = 30.

Crossrefs

Intersection of A290466 and A335142.

Programs

  • Mathematica
    zumQ[n_] := Module[{d = Divisors[n], ud, nd, sumUd, sumNd, x},ud = Select[d, CoprimeQ[#, n/#] &]; nd = Complement[d, ud]; sumUd = Plus @@ ud; sumNd = Plus @@ nd; sumUd >= 2*n && sumNd > 0 && EvenQ[sumUd] && EvenQ[sumNd] && CoefficientList[Product[1 + x^i, {i, ud}], x][[1 + sumUd/2]] > 0 && CoefficientList[Product[1 + x^i, {i, nd}], x][[1 + sumNd/2]] > 0]; Select[Range[10000], zumQ]
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.