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.

A321145 Exponential pseudoperfect numbers (A318100) equal to the sum of a subset of their proper exponential divisors 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, 4500, 4572, 4716, 4788, 4932
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2018

Keywords

Comments

The exponential version of A064771.

Examples

			4500 is in the sequence since its proper exponential divisors are 30, 60, 90, 180, 750, 1500, 2250 and {750, 1500, 2250} is the only subset that sums to 4500.

Links

Crossrefs

Cf. A064771, A295829, A295830, A318100.

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, #]&] ]; esigma[1]=1; esigma[n_] := Total@eDivs[n]; eDeficientQ[n_] := esigma[n] < 2n; a = {}; n = 0; While[Length[a] < 30, n++; If[eDeficientQ[n], Continue[]]; d = Most[eDivs[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c == 1, AppendTo[a, n]]]; a

A327946 Nonunitary pseudoperfect numbers (A327945) that equal to the sum of a subset of their nonunitary divisors in a single way.

Original entry on oeis.org

24, 36, 80, 112, 200, 312, 352, 392, 408, 416, 456, 552, 588, 684, 696, 744, 888, 984, 1032, 1088, 1116, 1128, 1216, 1272, 1332, 1416, 1464, 1472, 1548, 1608, 1692, 1704, 1752, 1856, 1896, 1908, 1936, 1984, 1992, 2124, 2136, 2196, 2288, 2328, 2412, 2424, 2472
Offset: 1

Views

Author

Amiram Eldar, Sep 30 2019

Keywords

Comments

The nonunitary version of A064771.

Examples

			The nonunitary divisors of 36 are {2, 3, 6, 12, 18}, and {6, 12, 18} is the only subset that sums to 36.

Crossrefs

Cf. A064771, A064591, A064597, A295829, A327945.

Programs

  • Mathematica
    nudiv[n_] := Module[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; s = {}; Do[d = nudiv[n]; If[Total[d] < n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c == 1, AppendTo[s, n]], {n, 1, 700}]; s

A335937 Infinitary pseudoperfect numbers (A306983) that equal to the sum of a subset of their aliquot infinitary divisors in a single way.

Original entry on oeis.org

6, 60, 72, 78, 88, 90, 96, 102, 104, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 426, 438, 474, 486, 498, 534, 582, 606, 618, 642, 654, 678, 726, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1014, 1038, 1074, 1086, 1146, 1158, 1182, 1194
Offset: 1

Views

Author

Amiram Eldar, Jun 30 2020

Keywords

Examples

			72 is a term since its set of infinitary aliquot divisors is {1, 2, 4, 8, 9, 18, 36}, and {1, 8, 9, 18, 36} is its only subset whose sum is equal to 72.

Links

Crossrefs

The infinitary version of A064771.
Subsequence of A306983.
A007357 is a subsequence.
Similar sequences: A295829, A295830, A321145.
Cf. A049417, A077609, A335199.

Programs

  • Mathematica
    idivs[x_] := If[x == 1, 1, Sort @ Flatten @ Outer[Times, Sequence @@ (FactorInteger[x] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; infpspQ[n_] := Module[{d = Most @ idivs[n], x}, Plus @@ d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] == 1]; Select[Range[2, 1200], infpspQ]
Showing 1-3 of 3 results.

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