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 1-3 of 3 results.

A083209 Numbers whose divisors can be partitioned in exactly one way into two disjoint sets with the same sum.

Original entry on oeis.org

6, 12, 20, 28, 56, 70, 88, 104, 176, 208, 272, 304, 368, 464, 496, 550, 650, 736, 836, 928, 992, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 2752, 3008, 3230, 3392, 3770, 3776, 3904, 4030, 4288, 4510, 4544, 4672, 5056, 5170, 5312, 5696, 5830, 6208, 6464
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 22 2003

Keywords

Comments

A083206(a(n))=1; perfect numbers (A000396) are a subset; problem: are weird numbers (A006037) a subset?
The weird numbers A006037 are not a subset of this sequence. The first missing weird number is A006037(8) = 10430. - Alois P. Heinz, Oct 29 2009
All numbers of the form p*2^k are in this sequence for k>0 and odd primes p between 2^(k+1)/3 and 2^(k+1). - T. D. Noe, Jul 08 2010
"Numbers with exactly one subset of their sets of divisors such that the complement has the same sum." - This was the original name of the sequence, but strictly taken is incorrect, because there are always two subsets that satisfy this condition: the subset and its complement. - Antti Karttunen, Dec 02 2024

Examples

			n=20: 2+4+5+10 = 1+20, 20 is a term (A083206(20)=1).

Links

Crossrefs

Subsequence of A083207, Zumkeller numbers.
Positions of 1's in A083206.
Cf. A005101, A005835, A064771, A337739 (terms with record number of divisors), A378449 (characteristic function), A378530 (subsequence).
Cf. also A378652, and A335143, A335199, A335202, A335219, A335217, A339980 for variants.

Programs

  • Maple
    with(numtheory): b:= proc(n,l) option remember; local m, ll, i; m:= nops(l); if n<0 then 0 elif n=0 then 1 elif m=0 or add(i, i=l)Alois P. Heinz, Oct 29 2009
  • Mathematica
    b[n_, l_] := b[n, l] = Module[{m, ll, i}, m = Length[l]; Which[n<0, 0, n == 0, 1, m == 0 || Total[l] Nothing]; b[n, ll] + b[n - l[[m]], ll]]]; a[n_] := a[n] = Module[{i, k, l, m, r}, For[k = If[n == 1, 1, a[n-1]+1], True, k++, l = Divisors[k]; {m, r} = QuotientRemainder[Total[l], 2]; If[r==0 && b[m, l]==2, Break[]]]; k]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 31 2017, after Alois P. Heinz *)
  • PARI
    isA083209 = A378449; \\ Antti Karttunen, Nov 28 2024

Extensions

More terms from Alois P. Heinz, Oct 29 2009
Improved the definition, old name moved to the comments - Antti Karttunen, Dec 02 2024

A339980 Coreful Zumkeller numbers (A339979) whose set of coreful divisors can be partitioned into two disjoint sets of equal sum in a single way.

Original entry on oeis.org

36, 72, 180, 200, 252, 360, 392, 396, 468, 504, 600, 612, 684, 784, 792, 828, 936, 1044, 1116, 1176, 1224, 1260, 1332, 1368, 1400, 1476, 1548, 1656, 1692, 1908, 1936, 1960, 1980, 2088, 2124, 2196, 2200, 2232, 2340, 2352, 2412, 2520, 2556, 2600, 2628, 2664, 2704
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 there is only one partition of its set of coreful divisors, {6, 12, 18, 36}, into 2 disjoint sets whose sums are equal: 6 + 12 + 18 = 36.

Crossrefs

A307958 is a subsequence.
Subsequence of A308053 and A339979.
Cf. A007947, A057723.
Similar sequences: A083209, A335143, A335199, A335202, A335217, A335219.

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]] == 2]; Select[Range[10000], corZumQ]

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
Showing 1-3 of 3 results.

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