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 1-10 of 12 results. Next

A348275 Odd noninfinitary abundant numbers: the odd terms of A348274.

Original entry on oeis.org

99225, 1091475, 1289925, 1334025, 1576575, 1686825, 1715175, 1863225, 1885275, 2027025, 2061675, 2282175, 2304225, 2395575, 2401245, 2436525, 2480625, 2650725, 2723175, 2789325, 2877525, 2962575, 3031875, 3075975, 3132675, 3185325, 3186225, 3296475, 3353805, 3501225
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Comments

The number of terms below 10^k, for k = 5, 6, ..., are 1, 113, 630, 7771, 73685, ... Apparently this sequence has an asymptotic density 0.000007...

Examples

			99225 is a term since A348271(99225) = 107207 > 99225.

Links

Crossrefs

Cf. A348271.
Subsequence of A005231 and A348274.
Similar sequences: A094889, A127666, A129485, A293186, A321147.

Programs

  • Mathematica
    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[1, 2*10^6, 2], s[#] > # &]

A348525 Noninfinitary weird numbers: noninfinitary abundant numbers (A348274) that are not equal to the sum of any subset of their noninfinitary divisors.

Original entry on oeis.org

3344, 12636, 88900, 95900, 109900, 116900, 121100, 181424, 271472, 365552, 476272, 504016, 975568, 1016048, 1354288, 1375504, 1407824, 1552304, 1628528, 1641904, 1862608, 1882672, 1902736, 1909424, 1929488, 1962928, 1982992, 2003056, 2009744, 2029808, 2049872
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Examples

			3344 is a term since the sum of its noninfinitary divisors, {2, 4, 8, 22, 38, 44, 76, 88, 152, 418, 836, 1672}, is 3360 > 3344, and no subset of these divisors sums to 3344.

Links

Crossrefs

Cf. A348271, A348274, A348341.
Similar sequences: A006037, A064114, A292986, A306984, A321146, A327948.

Programs

  • Mathematica
    q[n_] := !IntegerQ@ Log2@ DivisorSigma[0, n]; nidiv[1] = {}; nidiv[n_] := Complement[Divisors[n], Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; s = {}; Do[If[! q[n], Continue[]]; d = nidiv[n]; If[Total[d] <= n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c == 0, AppendTo[s, n]], {n, 1, 13000}]; s

Extensions

More terms from Amiram Eldar, Mar 25 2023

A348276 Numbers k such that k and k+1 are both noninfinitary abundant numbers (A348274).

Original entry on oeis.org

64198575, 84909824, 86424975, 110238975, 113223824, 191206575, 211266224, 224722575, 231058575, 231800624, 240069375, 240584175, 245383424, 262648575, 262911824, 279597824, 293893424, 297774224, 333773055, 338676975, 340250624, 340829775, 347244975, 372683024
Offset: 1

Views

Author

Amiram Eldar, Oct 09 2021

Keywords

Examples

			64198575 is a term since A348271(64198575) = 69470136 > 64198575 and A348271(64198576) = 65363424 > 64198576.

Crossrefs

Cf. A348271.
Subsequence of A096399 and A348274.
Similar sequences: A318167, A327635, A327942, A331412.

Programs

  • Mathematica
    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]; q[n_] := DivisorSigma[1,n] - isigma[n] > n; seq = {}; q1 = q[1]; Do[q2 = q[n]; If[q1 && q2, AppendTo[seq, n-1]]; q1=q2 ,{n,2,10^8}]; seq

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

Views

Author

Amiram Eldar, Oct 25 2021

Keywords

Comments

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...

Examples

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

Links

Crossrefs

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

Programs

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

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

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

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

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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]

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

Views

Author

Amiram Eldar, Dec 14 2024

Keywords

Comments

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... .

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • PARI
    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

Views

Author

Amiram Eldar, Oct 09 2022

Keywords

Comments

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... .

Examples

			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.

Links

Crossrefs

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

Programs

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

A348523 Numbers that are both infinitary and noninfinitary abundant numbers.

Original entry on oeis.org

960, 1440, 1800, 2016, 2400, 2940, 3240, 3528, 3780, 4536, 4860, 6720, 7260, 8640, 10080, 10140, 10560, 12096, 12480, 12600, 13860, 14784, 15120, 15360, 15840, 16320, 16380, 16800, 17472, 17640, 18240, 18480, 18720, 18900, 19008, 19800, 20160, 21420, 21600, 21840
Offset: 1

Views

Author

Amiram Eldar, Oct 21 2021

Keywords

Comments

Apparently, the smallest odd term is 9170790153525.

Examples

			960 is a term since A049417(960) = 2040 > 2*960 = 1920 and A348271(960) = 1008 > 960.

Links

Crossrefs

Intersection of A129656 and A348274.
Subsequence of A068403.
Cf. A049417, A348271.

Programs

  • Mathematica
    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]; q[n_] := (i = isigma[n]) > 2*n && DivisorSigma[1, n] - i > n; Select[Range[10^4], q]

A360525 Numbers k such that A360522(k) > 2*k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 180, 186, 204, 210, 222, 228, 246, 252, 258, 276, 282, 294, 300, 318, 330, 348, 354, 360, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564
Offset: 1

Views

Author

Amiram Eldar, Feb 10 2023

Keywords

Comments

First differs from A308127 at n = 15.
Analogous to abundant numbers (A005101) with A360522 instead of A000203.
Subsequence of A005101 because A360522(n) <= A000203(n) for all n.
The least odd term is a(1698) = A360526(1) = 15015.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 8, 95, 1135, 10890, 110867, 1104596, 11048123, 110534517, 1105167384, 11051009278, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1105...

Examples

			30 is a term since A360522(30) = 72 > 2*30.

Links

Crossrefs

Cf. A000203, A308127, A360522, A360524, A360526.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

Programs

  • Mathematica
    f[p_, e_] := p^e + e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := s[n] > 2*n; Select[Range[1000], q]
  • PARI
    is(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) > 2*n;}
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