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 1-4 of 4 results.

A335219 Exponential Zumkeller numbers (A335218) whose set of exponential divisors can be partitioned into two disjoint sets of equal sum in a single way.

Original entry on oeis.org

36, 180, 252, 396, 468, 612, 684, 828, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4572, 4716, 4788, 4900, 4932
Offset: 1

Views

Author

Amiram Eldar, May 27 2020

Keywords

Comments

Differs from A054979 first at a(44), since 4900 is in this sequence but not in A054979. - R. J. Mathar, Jun 02 2020

Examples

			36 is a term since there is a single way in which its exponential divisors, {6, 12, 18, 36} can be partitioned into 2 disjoint sets whose sum is equal: 6 + 12 + 18 = 36.

Links

Crossrefs

The exponential of A083209.
Subsequence of A335218.
Cf. A335143, A335199, A335202.

Programs

  • Mathematica
    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, #] &]]; ezQ[n_] := Module[{d = eDivs[n], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] == 2]]; Select[Range[10^4], ezQ]

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

Views

Author

Amiram Eldar, May 27 2020

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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

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[#] &]
Showing 1-4 of 4 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.