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.

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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

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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

A340110 Coreful 4-abundant numbers: numbers k such that csigma(k) > 4*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

10584000, 12700800, 15876000, 19051200, 21168000, 22226400, 25401600, 29635200, 31752000, 37044000, 38102400, 42336000, 44452800, 47628000, 50803200, 52920000, 55566000, 57153600, 59270400, 63504000, 64033200, 66679200, 74088000, 76204800, 79380000, 84672000
Offset: 1

Views

Author

Amiram Eldar, Dec 28 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).
Analogous to A068404 as A308053 is analogous to A005101.

Examples

			10584000 is a term since csigma(10584000) = 42653520 > 4 * 10584000.

Links

Crossrefs

Subsequence of A308053 and A340109.
Cf. A007947, A057723.
Similar sequences: A068404, A307114.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^8], s[#] > 4*# &]

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]

A340111 Coreful highly abundant numbers: numbers m such that csigma(m) > csigma(k) for all k < m, where csigma is the sum of the coreful divisors function (A057723).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 12, 16, 24, 32, 36, 48, 56, 64, 72, 96, 108, 128, 144, 192, 200, 216, 288, 360, 400, 432, 504, 576, 648, 720, 792, 800, 864, 1008, 1080, 1152, 1296, 1440, 1512, 1584, 1728, 1800, 1944, 2016, 2160, 2304, 2592, 2880, 3024, 3240, 3456, 3600
Offset: 1

Views

Author

Amiram Eldar, Dec 28 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).
Analogous to highly abundant numbers (A002093) with the sum of the coreful divisors function (A057723) instead of the sum of divisors function (A000203).

Examples

			The first 10 values of A057723(n) for n=1..10 are: 1, 2, 3, 6, 5, 6, 7, 14, 12, 10. The record values, 1, 2, 3, 6, 7 and 14 occur at 1, 2, 3, 4, 7 and 8, the first 6 terms of this sequence.

Links

Crossrefs

Cf. A007947, A057723, A283052, A308053.
Similar sequences: A002093, A285614, A292983, A327634, A328134, A329883.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 3600}]; seq
Previous Showing 11-14 of 14 results.

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