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 1-5 of 5 results.

A323342 Numbers k whose bi-unitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

704, 1458, 2394, 7544, 10184, 46400, 60416, 106434, 115182, 118098, 121014, 125000, 129762, 141426, 147258, 150174, 156006, 158922, 164754, 176418, 185166, 190998, 199746, 202662, 217242, 220158, 228906, 237654, 243486, 246402, 252234, 260982, 263898, 278478
Offset: 1

Views

Author

Amiram Eldar, Jan 11 2019

Keywords

Comments

The bi-unitary version of A171641.

Links

Crossrefs

Cf. A171641, A188999, A292982, A323341, A323343, A323344.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[n_] := Select[Divisors[n], Last@Intersection[f@#, f[n/#]] == 1 &]; fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[n_] := If[n==1, 1, Times @@ (fun @@@ FactorInteger[n])]; seq={}; Do[s=bsigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = bdiv[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 10000}]; seq

A323343 Numbers k whose exponential divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

1910412, 9552060, 21014532, 24835356, 32477004, 43939476, 55401948, 59222772, 70685244, 78326892, 82147716, 89789364, 101251836, 105072660, 112714308, 116535132, 124176780, 127997604, 135639252, 139460076, 150922548, 158564196, 162385020, 170026668, 185309964
Offset: 1

Views

Author

Amiram Eldar, Jan 11 2019

Keywords

Comments

The exponential version of A171641.

Links

Crossrefs

Cf. A051377, A129575, A171641, A323341, A323342, A323344.

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]]]} ]]; ediv[n_] := Module[ {d=Rest[Divisors[n]]}, Select[ d, expDivQ[n, #]&]]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[ Last[#]]}] &) /@ FactorInteger[n]; seq={}; Do[s=esigma[n]; If[OddQ[s] || s<=2n, Continue[]]; div = ediv[n]; If[Coefficient[Times @@ (1 + x^div) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 10000}]; seq

A335197 Infinitary Zumkeller numbers: numbers whose set of infinitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 54, 56, 60, 66, 70, 72, 78, 88, 90, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498, 504
Offset: 1

Views

Author

Amiram Eldar, May 26 2020

Keywords

Examples

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

Links

Crossrefs

The infinitary version of A083207.
Subsequence of A129656.
Cf. A077609, A323344.

Programs

  • Mathematica
    infdivs[n_] := If[n == 1, {1}, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infZumQ[n_] := Module[{d = infdivs[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[500], infZumQ] (* after Michael De Vlieger at A077609 *)

A323341 Numbers k whose unitary divisors have an even sum which is larger than 2k, but they cannot be partitioned into two disjoint parts whose sums are equal.

Original entry on oeis.org

2394, 1452330, 5771934, 5786298, 5800662, 5834178, 5843754, 5858118, 5886846, 5905998, 5920362, 5929938, 5992182, 6035274, 6059214, 6078366, 6087942, 6102306, 6107094, 6121458, 6174126, 6202854, 6207642, 6245946, 6265098, 6274674, 6303402, 6336918, 6360858
Offset: 1

Views

Author

Amiram Eldar, Jan 11 2019

Keywords

Links

Crossrefs

The unitary version of A171641.
Cf. A034448, A034683, A290466, A323342, A323343, A323344.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; seq={}; Do[s=usigma[n]; If[OddQ[s] || s<=2n, Continue[]]; udiv = Select[Divisors[n], GCD[ #, n/# ] == 1 &]; If[Coefficient[Times @@ (1 + x^udiv) // Expand, x, s/2] == 0, AppendTo[seq, n]], {n, 1, 1500000}]; seq

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]
Showing 1-5 of 5 results.

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