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.

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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).

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

Views

Author

Amiram Eldar, Apr 10 2018

Keywords

Examples

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

Links

Crossrefs

The bi-unitary version of A071927.
Cf. A188999, A292982, A302570.

Programs

  • Mathematica
    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
  • PARI
    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

Views

Author

Amiram Eldar, May 16 2020

Keywords

Comments

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).

Links

Crossrefs

The bi-unitary version of A005153.
Cf. A004171, A188999, A286652, A292982, A292985, A334901.

Programs

  • Mathematica
    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]

A307161 Numbers n such that A307159(n) = Sum_{k=1..n} bsigma(k) is divisible by n, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

1, 2, 17, 37, 50, 56, 391, 919, 1399, 2829, 6249, 13664, 28829, 62272, 67195, 585391, 5504271, 6798541, 10763933, 866660818, 3830393407, 11044287758, 23058607363, 83159875881, 206501883259, 297734985607, 1087473543732, 1184060078117, 2789730557061, 2821551579466, 3529184155643
Offset: 1

Views

Author

Amiram Eldar, Mar 27 2019

Keywords

Comments

The bi-unitary version of A056550.
The corresponding quotients are 1, 2, 13, 28, 38, 43, ... (see the link for more values).
a(32) > 10^13. - Giovanni Resta, May 28 2019

Links

Crossrefs

Cf. A056550, A064611, A188999, A307043, A307159.

Programs

  • Mathematica
    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]); seq={};s = 0; Do[s = s + bsigma[n]; If[Divisible[s,n], AppendTo[seq,n]], {n, 1, 10^6}]; seq

Extensions

a(23)-a(31) from Giovanni Resta, Apr 20 2019

A379615 Numerators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 4, 19, 107, 39, 61, 259, 89, 93, 857, 887, 181, 1303, 331, 1345, 4091, 4175, 21127, 4301, 21757, 87973, 88813, 90073, 90577, 1192621, 1201981, 1211809, 1221637, 1234741, 1240201, 626243, 89909, 45247, 15169, 30533, 153601, 2941819, 2956639, 20807623, 20876783
Offset: 1

Views

Author

Amiram Eldar, Dec 27 2024

Keywords

Examples

			Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 89/40, 93/40, 857/360, 887/360, 181/72, ...

Links

Crossrefs

Cf. A188999, A307159, A370904, A379616 (denominators), A379617.

Programs

  • Mathematica
    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]; Numerator[Accumulate[Table[1/bsigma[n], {n, 1, 50}]]]
  • PARI
    bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / bsigma(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} 1/A188999(k)).
a(n)/A379616(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) / n), where A and B are constants.

A379616 Denominators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 3, 12, 60, 20, 30, 120, 40, 40, 360, 360, 72, 504, 126, 504, 1512, 1512, 7560, 1512, 7560, 30240, 30240, 30240, 30240, 393120, 393120, 393120, 393120, 393120, 393120, 196560, 28080, 14040, 4680, 9360, 46800, 889200, 889200, 6224400, 6224400, 889200, 1778400
Offset: 1

Views

Author

Amiram Eldar, Dec 27 2024

Keywords

Links

Crossrefs

Cf. A188999, A307159, A370904, A379615 (numerators), A379618.

Programs

  • Mathematica
    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]; Denominator[Accumulate[Table[1/bsigma[n], {n, 1, 50}]]]
  • PARI
    bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / bsigma(k); print1(denominator(s), ", "))};

Formula

a(n) = denominator(Sum_{k=1..n} 1/A188999(k)).

A379617 Numerators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 2, 11, 43, 53, 4, 37, 103, 23, 65, 71, 337, 2539, 1217, 2539, 7337, 7757, 1501, 7883, 7631, 31469, 30629, 31889, 6277, 84625, 82753, 423593, 82753, 426869, 421409, 216847, 213727, 108911, 11899, 24253, 119081, 2317139, 760853, 773203, 6889667, 7037867, 13946059
Offset: 1

Views

Author

Amiram Eldar, Dec 27 2024

Keywords

Examples

			Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 103/120, 23/24, 65/72, 71/72, 337/360, ...

Links

Crossrefs

Cf. A188999, A307159, A370904, A379615, A379618 (denominators).

Programs

  • Mathematica
    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]; Numerator[Accumulate[Table[(-1)^(n+1)/bsigma[n], {n, 1, 50}]]]
  • PARI
    bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / bsigma(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A188999(k)).
a(n)/A379618(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) * n^c), where c = log(9/10)/log(2) = -0.152003..., and A and B are constants.

A379618 Denominators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

Original entry on oeis.org

1, 3, 12, 60, 60, 5, 40, 120, 24, 72, 72, 360, 2520, 1260, 2520, 7560, 7560, 1512, 7560, 7560, 30240, 30240, 30240, 6048, 78624, 78624, 393120, 78624, 393120, 393120, 196560, 196560, 98280, 10920, 21840, 109200, 2074800, 691600, 691600, 6224400, 6224400, 12448800
Offset: 1

Views

Author

Amiram Eldar, Dec 27 2024

Keywords

Links

Crossrefs

Cf. A188999, A307159, A370904, A379616, A379617 (numerators).

Programs

  • Mathematica
    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]; Denominator[Accumulate[Table[(-1)^(n+1)/bsigma[n], {n, 1, 50}]]]
  • PARI
    bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / bsigma(k); print1(denominator(s), ", "))};

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A188999(k)).

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

Views

Author

Amiram Eldar and Michael De Vlieger, Apr 22 2018

Keywords

Comments

The bi-unitary version of A271816.
Includes all the odd powers of 2 (A004171).

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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
  • PARI
    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

Views

Author

Amiram Eldar and Michael De Vlieger, Apr 22 2018

Keywords

Comments

The bi-unitary version of A181595.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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
  • PARI
    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

A369204 Numbers m such that A034448(A188999(m)) = k*m for some k, where A034448 and A188999 are respectively the unitary and the bi-unitary sigma function.

Original entry on oeis.org

1, 2, 8, 9, 10, 18, 24, 27, 30, 54, 165, 238, 288, 512, 656, 660, 864, 952, 1536, 1968, 2464, 2880, 4608, 4680, 13824, 14448, 14976, 16728, 19008, 19992, 23040, 29376, 60928, 152064, 155520, 172368, 279552, 474936, 746928, 1070592, 1114560, 1524096, 1703520
Offset: 1

Views

Author

Tomohiro Yamada, Jan 16 2024

Keywords

Examples

			A188999(18) = 4 * 10 = 40 and A034448(40) = 9 * 6 = 54 = 3 * 18, so 18 is a term with k = 3.

Links

Crossrefs

Cf. A038843 (analog for A034448(A034448(m))), A318175 (analog for A188999(A188999(m))).
Cf. A369205 (analog for A188999(A034448(m))).

Programs

  • PARI
    a034448(n) = {my(f,i,p,e);f=factor(n);for(i=1,#f~,p=f[i,1];e=f[i,2];f[i,1]=p^e+1;f[i,2]=1);factorback(f)};
a188999(n) = {my(f,i,p,e);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) = (a034448(a188999(n))%n) == 0;
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