A292982 -id:A292982 - cp's OEIS frontend

A292986 Bi-unitary weird numbers: bi-unitary abundant numbers (A292982) that are not bi-unitary pseudoperfect (A292985).

70, 4030, 5390, 5830, 7192, 7400, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 11830, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810

Offset: 1

Amiram Eldar, Sep 27 2017

nonn

Analogous to weird numbers (A006037) with bi-unitary sigma (A188999) instead of sigma (A000203).

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

Cf. A006037, A064114, A188999, A292982, A292985.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 2 n; a = {}; n = 0; While[Length[a] < 5, n++; If[!bAbundantQ[n], Continue[]]; d = Most[bdiv[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c <= 0, AppendTo[a, n]]]; a (* after T. D. Noe at A005835 and Michael De Vlieger at A188999 *)

A358484 Numbers k such that k, k+1 and k+2 are all bi-unitary abundant numbers (A292982).

Original entry on oeis.org

268005374, 600350750, 2666847104, 2683146464, 2695309694, 2849458688, 3904592768, 4112553248, 5368737374, 6554410784, 6955574624, 8207456894, 8967010688, 9220179968, 9868366430, 10529171288, 12147283070, 12411630944, 12491149670, 13911605630, 14126720894, 14396391008

Offset: 1

Amiram Eldar, Nov 18 2022

nonn

			268005374 is in the sequence since 268005374, 268005375 and 268005376 are all bi-unitary abundant numbers (A292982): bsigma(268005374) = 568995840 > 2 * 268005374, bsigma(268005375) = 540633600 > 2 * 268005375, and bsigma(268005376) = 541258200 > 2 * 268005376 (bsigma is the sum of bi-unitary divisors, A188999).

Amiram Eldar, Table of n, a(n) for n = 1..1546 (terms below 10^12)

Subsequence of A096536, A292982 and A318167.

Cf. A188999.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; abQ[n_] := bsigma[n] > 2*n; v = Cases[Import["https://oeis.org/A096536/b096536.txt", "Table"], {, }][[;; , 2]]; Select[v, And @@ abQ /@ (# + {0, 1, 2}) &]

A293186 Odd bi-unitary abundant numbers: odd numbers k such that bsigma(k) > 2*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

945, 8505, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25515, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31185, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 36855, 37125, 38745, 39585, 40635, 41055

Offset: 1

Amiram Eldar, Oct 01 2017

nonn

Analogous to odd abundant numbers (A005231) with bi-unitary sigma (A188999) instead of sigma (A000203).

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 1, 2, 82, 559, 6493, 61831, 642468, 6339347, 63112602, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00063... . - Amiram Eldar, Sep 02 2022

			945 is in the sequence since bsigma(945) = 1920 > 2*945.

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

Cf. A005231, A129485, A188999, A292982.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=
DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bOddAbundantQ[n_] := OddQ[n] && bsigma[n] > 2 n; Select[Range[1000], bOddAbundantQ] (* after Michael De Vlieger at A188999 *)

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));
biusig(n) = vecsum(biudivs(n));
isok(n) = (n % 2) && (biusig(n) > 2*n); \\ Michel Marcus, Dec 15 2017

A348274 Noninfinitary abundant numbers: numbers k such that A348271(k) > k.

Original entry on oeis.org

36, 48, 80, 144, 180, 240, 252, 288, 300, 324, 336, 396, 400, 432, 468, 528, 560, 576, 588, 612, 624, 684, 720, 768, 784, 816, 828, 880, 900, 912, 960, 1008, 1040, 1044, 1104, 1116, 1200, 1232, 1260, 1280, 1296, 1332, 1360, 1392, 1440, 1456, 1476, 1488, 1520, 1548

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

The first odd term is a(3577) = 99225.

The number of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 3, 31, 360, 3605, 36160, 360840, 3618980, 36144059, ... Apparently this sequence has an asymptotic density 0.0361...

			36 is a term since A348271(36) = 41 > 36.

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

Cf. A348271, A327633.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1500], s[#] > # &]

A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

Original entry on oeis.org

24, 30, 42, 48, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 114, 120, 126, 132, 138, 150, 156, 160, 162, 168, 174, 180, 186, 192, 198, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 300, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The smallest odd term is a(1357) = 8505.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 13, 148, 1595, 15688, 158068, 1578957, 15762209, 157745113, 1577808429, ... Apparently this sequence has an asymptotic density 0.157...

			24 is a term since A160135(24) = 30 > 24.

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

Cf. A160135, A348601.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[400], q]

A318167 Numbers k such that both k and k+1 are bi-unitary abundant numbers.

Original entry on oeis.org

21735, 21944, 43064, 49664, 58695, 76544, 106784, 135135, 144584, 160544, 188055, 209055, 227744, 256095, 262184, 300104, 345344, 348704, 382304, 387584, 407295, 409184, 414855, 437535, 498015, 520695, 560384, 567944, 611415, 679455, 687015, 705375, 709695

Offset: 1

Amiram Eldar, Aug 20 2018

nonn

The bi-unitary version of A096399.

			21735 is in the sequence since both 21735 and 21736 are bi-unitary abundant numbers.

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

Cf. A096399 (analog for sigma), A188999 (bi-unitary sigma).

Cf. A292982 (bi-unitary abundant), A293186 (odd bi-unitary abundant).

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] > 2 n; seq={}; n=1; While[Length[seq]<32,If[bAbundantQ[n] && bAbundantQ [n+1],AppendTo[seq,n]];n++];seq

a188999(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
isok(n) = (a188999(n) > 2*n) && (a188999(n+1) > 2*(n+1)); \\ Michel Marcus, Aug 21 2018

A380929 Numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

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

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

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

Amiram Eldar, Jan 11 2019

nonn

The bi-unitary version of A171641.

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

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

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

A302994 Number of bi-unitary abundant numbers < 10^n.

Original entry on oeis.org

0, 14, 147, 1553, 15450, 155395, 1549818, 15498814, 155079196, 1550331185, 15503061466, 155037242668, 1550370696100, 15503650949671, 155036854371220, 1550366484701654, 15503648102080675

Offset: 1

Amiram Eldar, Apr 17 2018

Cf. A188999, A292982, A302992.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=
DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; babQ[n_] := bsigma[n] > 2 n; c = 0; k = 1; seq={}; Do[While[k < 10^n, If[babQ[k], c++]; k++]; AppendTo[seq, c], {n, 1, 5}]; seq

biusigma(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
a(n) = sum(k=1, 10^n-1, biusigma(k) > 2*k); \\ Michel Marcus, Apr 17 2018

Conjecture: Lim_{n->oo} a(n)/10^n = 0.15... is the density of bi-unitary abundant numbers.

a(8)-a(17) from Hiroaki Yamanouchi, Aug 24 2018

A333928 Recursive abundant numbers: numbers k such that A333926(k) > 2*k.

Original entry on oeis.org

12, 18, 20, 30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 108, 114, 120, 126, 132, 138, 140, 144, 150, 156, 168, 174, 180, 186, 196, 198, 204, 210, 220, 222, 228, 234, 240, 246, 252, 258, 260, 270, 276, 282, 294, 300, 306, 308, 318, 324, 330, 336, 340, 342, 348

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

			12 is a term since A333926(12) = 28 > 2 * 12.

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

Cf. A333926, A333927.

Analogous sequences: A005101, A034683 (unitary), A064597 (nonunitary), A129575 (exponential), A129656 (infinitary), A292982 (bi-unitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[350], recDivSum[#] > 2*# &]

cp's OEIS Frontend

A292986 Bi-unitary weird numbers: bi-unitary abundant numbers (A292982) that are not bi-unitary pseudoperfect (A292985).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A358484 Numbers k such that k, k+1 and k+2 are all bi-unitary abundant numbers (A292982).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A293186 Odd bi-unitary abundant numbers: odd numbers k such that bsigma(k) > 2*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A348274 Noninfinitary abundant numbers: numbers k such that A348271(k) > k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A318167 Numbers k such that both k and k+1 are bi-unitary abundant numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A380929 Numbers k such that A380845(k) > 2*k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs