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.

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

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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 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]

A339981 Primitive coreful Zumkeller numbers: coreful Zumkeller numbers (A339979) having no coreful Zumkeller aliquot divisor.

Original entry on oeis.org

36, 200, 392, 1936, 2704, 4900, 9248, 11552, 16928, 26912, 30752, 60500, 84500, 87616, 99225, 107584, 118336, 141376, 163592, 165375, 179776, 222784, 231525, 238144, 349448, 574592, 645248, 682112, 798848, 881792, 1013888, 1204352, 1225125, 1305728, 1357952
Offset: 1

Views

Author

Amiram Eldar, Dec 25 2020

Keywords

Comments

If m is a coreful Zumkeller number and k is a squarefree number such that gcd(m, k) = 1, then k*m is also a coreful Zumkeller number.

Examples

			a(1) = 36 since it is the least coreful Zumkeller number.
The next coreful Zumkeller numbers, 72, 144 and 180, are not terms since they are multiples of 36.

Crossrefs

Subsequence of A339979.
A307959 is a subsequence.
Similar sequence: A180332.

Programs

  • Mathematica
    corZumQ[n_] := 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]; primczQ[n_] := corZumQ[n] && NoneTrue[Most @ Divisors[n], corZumQ]; Select[Range[10^6], primczQ]

A339983 Coreful abundant numbers (A308053) with an even sum of coreful divisors (A057723) that are not coreful Zumkeller numbers (A339979).

Original entry on oeis.org

108, 216, 432, 540, 756, 864, 972, 1000, 1080, 1188, 1404, 1512, 1728, 1836, 1944, 2000, 2052, 2160, 2376, 2484, 2744, 2808, 3000, 3024, 3132, 3348, 3456, 3672, 3780, 3888, 3996, 4000, 4104, 4320, 4428, 4644, 4752, 4860, 4968, 5076, 5488, 5616, 5724, 5940, 6000
Offset: 1

Views

Author

Amiram Eldar, Dec 25 2020

Keywords

Examples

			108 is a term since its set of coreful divisors, {6, 12, 18, 36, 54, 108}, has an even sum, 234 > 2*108, and it cannot be partitioned into two disjoint sets of equal sum.

Crossrefs

Subsequence of A308053.
Cf. A057723, A339979, A339982.
Similar sequences: A171641, A323341, A323342, A323343, A323344.

Programs

  • Mathematica
    q[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[10^4], q]

A339982 Coreful abundant numbers (A308053) with an odd sum of coreful divisors.

Original entry on oeis.org

1157625, 10418625, 12733875, 15049125, 19679625, 21994875, 26625375, 28940625, 33571125, 35886375, 40429125, 42832125, 47462625, 49777875, 54408375, 56723625, 61354125, 66733875, 68299875, 70615125, 77560875, 82191375, 84506625, 91452375, 93767625, 96082875
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).
All the terms are odd numbers since the sum of coreful divisors (A057723) of an even number is even.
All the terms are exponentially odd numbers (A268335) since the sum of coreful divisors function is multiplicative and A057723(p^e) = p + p^2 + ... + p^e is even for a prime p and an even exponent e.
None of the terms are coreful Zumkeller numbers (A339979).

Examples

			1157625 is a term since A057723(1157625) = 2411955 > 2*1157625 and it is odd.

Links

Crossrefs

Intersection of A268335 and A339936.
Subsequence of A308053.
Cf. A007947, A057723, A339979.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1, 2*10^7, 2], (sum = s[#]) > 2*# && OddQ[sum] &]
Showing 1-4 of 4 results.

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