A292982 -id:A292982 - cp's OEIS frontend

A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2d = 2k, where bsigma is the sum of bi-unitary divisors function (A188999).

24, 30, 40, 42, 48, 54, 56, 66, 70, 78, 80, 88, 102, 104, 114, 120, 138, 150, 162, 174, 186, 222, 224, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 448, 474, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 672, 678, 720, 726, 762, 780, 786

Offset: 1

Amiram Eldar, May 18 2020

nonn

Equivalently, numbers that are equal to the sum of their proper bi-unitary divisors, with one of them taken with a minus sign.

Admirable numbers (A111592) that are exponentially odd (A268335) are also bi-unitary admirable numbers since all of their divisors are bi-unitary. Terms that are not exponentially odd are 48, 80, 150, 162, 294, 360, 420, 448, 540, 630, 660, 720, 726, 780, 832, 990, ...

			48 is in the sequence since 48 = 1 + 2 + 3 - 6 + 8 + 16 + 24 is the sum of its proper bi-unitary divisors with one of them, 6, taken with a minus sign.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The bi-unitary version of A111592.
Subsequence of A292982.
Cf. A188999, A222266, A268335, A328328, A334974.

fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); buDivQ[n_, 1] = True; buDivQ[n_, div_] := If[Mod[#2, #1] == 0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; buAdmQ[n_] := (ab = bsigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && buDivQ[n, ab/2]; Select[Range[1000], buAdmQ]

A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

6, 24, 30, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 88, 90, 96, 102, 104, 114, 120, 138, 150, 160, 162, 168, 174, 186, 192, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390, 402

Offset: 1

Amiram Eldar, May 27 2020

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

The bi-unitary version of A083207.
Subsequence of A292982.
Cf. A188999, A222266, A290466, A323342, A335142, A335197.

uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[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[10^3], bzQ]

A357605 Numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

36, 48, 72, 80, 96, 108, 120, 144, 160, 162, 168, 180, 192, 200, 216, 224, 240, 252, 264, 270, 280, 288, 300, 312, 320, 324, 336, 352, 360, 378, 384, 392, 396, 400, 408, 416, 432, 448, 450, 456, 468, 480, 486, 500, 504, 528, 540, 552, 560, 576, 588, 594, 600, 612

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

The least odd term is a(470) = A357607(1) = 4725.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 5, 92, 1011, 10160, 102125, 1022881, 10231151, 102249758, 1022781199, 10229781638, ... . Apparently, the asymptotic density of this sequence exists and equals 0.102... .
An analog of abundant numbers, in which the divisor sum is restricted to nonsquarefree divisors. - Peter Munn, Oct 26 2022

			36 is a term since A162296(36) = 79 > 2*36.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A162296.
Subsequence of A005101 and A013929.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[2, 1000], q]

A293187 Bi-unitary 3-abundant numbers: numbers k such that bsigma(k) > 3*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

480, 840, 1080, 1320, 1512, 1560, 1680, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3240, 3360, 3480, 3720, 3840, 4320, 4440, 4920, 5160, 5280, 5640, 5880, 6048, 6240, 6360, 6720, 7080, 7320, 7392, 7560, 7680, 8040, 8160, 8520

Offset: 1

Amiram Eldar, Oct 01 2017

nonn

Analogous to 3-abundant numbers (A023197) with bi-unitary sigma (A188999) instead of sigma (A000203).

			480 is in the sequence since bi-unitary sigma(480) = 1512 > 3 * 480.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023197, A285615, A188999, A292982.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=  DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 3 n; Select[Range[1000], bAbundantQ] (* after Michael De Vlieger at A188999 *)

A302571 Bi-unitary barely abundant numbers: bi-unitary abundant numbers k such that bsigma(k)/k < bsigma(m)/m for all bi-unitary abundant numbers m < k, where bsigma(k) is the sum of the bi-unitary divisors of k (A188999).

Original entry on oeis.org

24, 30, 40, 54, 56, 70, 80, 104, 642, 654, 678, 726, 762, 786, 822, 832, 1888, 1952, 4030, 5830, 7424, 32128, 62464, 374802, 374838, 374862, 374898, 374982, 375006, 375042, 375198, 375234, 375294, 375378, 375486, 375546, 375582, 375618, 375702, 375762, 375798

Offset: 1

Amiram Eldar, Apr 10 2018

nonn

			The values of bsigma(k)/k are: 3, 2.5, 2.4, 2.25, 2.222..., 2.142...

Amiram Eldar, Table of n, a(n) for n = 1..10000

The bi-unitary version of A071927.

Cf. A188999, A292982, A302570.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=  DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; r = 3; seq={}; Do[
s = bsigma[n]/n; If[s > 2 && s < r, AppendTo[seq,n]; r = s], {n, 1, 10000}]; seq

babindex(n) = {my(f = factor(n), p, e); prod(k = 1, #f~, p = f[k, 1]; e = f[k, 2]; (p^(e+1)-1)/(p^(e+1)-p^e) - if(e%2, 0, 1/p^(e/2)));}
lista(kmax) = {my(bab, babm = 3); for(k = 1, kmax, bab = babindex(k); if(bab > 2 && bab < babm, babm = bab; print1(k, ", "))); }

A334898 Bi-unitary practical numbers: numbers m such that every number 1 <= k <= bsigma(m) is a sum of distinct bi-unitary divisors of m, where bsigma is A188999.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 32, 40, 42, 48, 54, 56, 66, 72, 78, 88, 96, 104, 120, 128, 160, 168, 192, 210, 216, 224, 240, 264, 270, 280, 288, 312, 320, 330, 336, 352, 360, 378, 384, 390, 408, 416, 432, 440, 448, 456, 462, 480, 486, 504, 510, 512, 520, 528, 544, 546, 552

Offset: 1

Amiram Eldar, May 16 2020

nonn

Includes 1 and all the odd powers of 2 (A004171). The other terms are a subset of bi-unitary abundant numbers (A292982) and bi-unitary pseudoperfect numbers (A292985).

Amiram Eldar, Table of n, a(n) for n = 1..2500

The bi-unitary version of A005153.

Cf. A004171, A188999, A286652, A292982, A292985, A334901.

biunitaryDivisorQ[div_, n_] := If[Mod[#2, #1] == 0, Last @ Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; bdivs[n_] := Module[{d = Divisors[n]}, Select[d, biunitaryDivisorQ[#, n] &]]; bPracQ[n_] := Module[{d = bdivs[n], sd, x}, sd = Plus @@ d; Min @ CoefficientList[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, sd}], x] >  0]; Select[Range[1000], bPracQ]

A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 120, 138, 150, 168, 174, 186, 210, 222, 246, 258, 270, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822

Offset: 1

Amiram Eldar, Dec 14 2024

All the squarefree abundant numbers (A087248) are terms since A241405(k) = A000203(k) for a squarefree number k.
If k is a term and m is coprime to k them k*m is also a term.
The numbers of terms that do no exceed 10^k, for k = 2, 3, ..., are 5, 67, 767, 7595, 76581, 764321, 7644328, 76468851, 764630276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07646... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A241405, A323757, A323758, A323759, A379027.
Subsequence of A005101.
Subsequences: A034683, A087248, A379030, A379031.
Similar sequences: A064597, A129575, A129656, A292982, A348274, A348604.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1000], meAbQ]

is(n) = {my(f=factor(n)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*n;}

A357685 Numbers k such that A293228(k) > k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 102, 114, 132, 138, 140, 156, 174, 186, 204, 210, 222, 228, 246, 258, 276, 282, 318, 330, 348, 354, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564, 570, 582, 606, 618, 636, 642, 654, 660, 678, 690

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 7, 79, 843, 8230, 83005, 826875, 8275895, 82790525, 827718858, 8276571394, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0827... .

			30 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15} and their sum is 42 > 30.
60 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A293228.
Disjoint union of A087248 and A357686.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

s[n_] := Times @@ (1 + (f = FactorInteger[n])[[;; , 1]]) - If[AllTrue[f[[;;, 2]], # == 1 &], n, 0]; Select[Range[2, 1000], s[#] > # &]

is(n) = {my(f = factor(n), s); s = prod(i=1, #f~, f[i,1]+1); if(n==1 || vecmax(f[,2]) == 1, s -= n); s > n};

A303358 Bi-unitary deficient-perfect numbers: bi-unitary deficient numbers k for such that 2*k - bsigma(k) is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

1, 2, 8, 10, 12, 32, 112, 128, 136, 144, 152, 184, 512, 1088, 2048, 2144, 2272, 2528, 2736, 3248, 3312, 4592, 7936, 8192, 9800, 11800, 17176, 18632, 18904, 22984, 32768, 32896, 33664, 34688, 49024, 57152, 77248, 85952, 131072, 176400, 212400, 309168, 335376

Offset: 1

Amiram Eldar and Michael De Vlieger, Apr 22 2018

nonn

The bi-unitary version of A271816.

Includes all the odd powers of 2 (A004171).

			112 is in the sequence since the sum of its bi-unitary divisors is 1 + 2 + 7 + 8 + 14 + 16 + 56 + 112 = 216 and 2*112 - 216 = 8 is a bi-unitary divisor of 112.

Amiram Eldar, Table of n, a(n) for n = 1..171

Cf. A004171, A188999, A271816, A292982, A303359.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; biunitaryDivisorQ[ div_, n_] := If[Mod[#2,#1]==0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]]&, {#1, #2/#1}]]==1, False]& @@{div, n}; aQ[n_] := Module[{d=2n-bsigma[n]},If[d<=0, False,biunitaryDivisorQ[d,n]]]; s={}; Do[ If[aQ[n], AppendTo[s,n]], {n, 1, 10000}]; s

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
isok(n) = my(divs = biudivs(n), sig = vecsum(divs)); (sig < 2*n) && vecsearch(divs, 2*n-sig); \\ Michel Marcus, Apr 27 2018

A303359 Bi-unitary near-perfect numbers: bi-unitary abundant numbers k such that the abundance d = bsigma(k) - 2*k is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

24, 40, 56, 80, 88, 104, 120, 224, 360, 432, 672, 832, 992, 1008, 1296, 1456, 1504, 1584, 1888, 1952, 2016, 2160, 2800, 3800, 5624, 5800, 7424, 7616, 9112, 10080, 11096, 13736, 15872, 16256, 17816, 22848, 24448, 28544, 30592, 32128, 33728, 51136, 62464, 66368

Offset: 1

Amiram Eldar and Michael De Vlieger, Apr 22 2018

nonn

The bi-unitary version of A181595.

			24 is in the sequence since the sum of its bi-unitary divisors is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and 60 - 2*24 = 12 is a bi-unitary divisor of 24.

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A153501, A181595, A188999, A292982, A303358.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; biunitaryDivisorQ[ div_, n_] := If[Mod[#2, #1]==0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]]&, {#1, #2/#1}]]==1, False]& @@{div, n}; aQ[n_] := Module[{d=bsigma[n]-2n},If[d<=0, False,biunitaryDivisorQ[d,n]]]; s={}; Do[If[ aQ[n], AppendTo[s,n] ], {n, 1, 10000}]; s

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
isok(n) = my(divs = biudivs(n), sig = vecsum(divs)); (sig > 2*n) && vecsearch(divs, sig - 2*n); \\ Michel Marcus, Apr 27 2018

cp's OEIS Frontend

A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2*d = 2*k, where bsigma is the sum of bi-unitary divisors function (A188999).

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A335215 Bi-unitary Zumkeller numbers: numbers whose set of bi-unitary divisors can be partitioned into two disjoint sets of equal sum.

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A357605 Numbers k such that A162296(k) > 2*k.

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A293187 Bi-unitary 3-abundant numbers: numbers k such that bsigma(k) > 3*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

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A302571 Bi-unitary barely abundant numbers: bi-unitary abundant numbers k such that bsigma(k)/k < bsigma(m)/m for all bi-unitary abundant numbers m < k, where bsigma(k) is the sum of the bi-unitary divisors of k (A188999).

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A334898 Bi-unitary practical numbers: numbers m such that every number 1 <= k <= bsigma(m) is a sum of distinct bi-unitary divisors of m, where bsigma is A188999.

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A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

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A357685 Numbers k such that A293228(k) > k.

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A334972 Bi-unitary admirable numbers: numbers k such that there is a proper bi-unitary divisor d of k such that bsigma(k) - 2d = 2k, where bsigma is the sum of bi-unitary divisors function (A188999).