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.

Previous Showing 11-18 of 18 results.

A339936 Odd coreful abundant numbers: the odd terms of A308053.

Original entry on oeis.org

99225, 165375, 231525, 297675, 496125, 694575, 826875, 893025, 1091475, 1157625, 1225125, 1289925, 1488375, 1620675, 1686825, 1819125, 1885275, 2083725, 2149875, 2282175, 2480625, 2546775, 2679075, 2811375, 2877525, 3009825, 3075975, 3142125, 3274425, 3472875
Offset: 1

Views

Author

Amiram Eldar, Dec 23 2020

Keywords

Comments

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 9.1348...*10^(-6), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

Examples

			99225 is a term since it is odd and the sum of its coreful divisors is A057723(99225) = 201600 > 2 * 99225.

Links

Crossrefs

Intersection of A005408 and A308053.
Subsequence of A321147.
Cf. A057723, A356871.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A293186, A321147, A333950.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1, 10^6, 2], s[#] > 2*# &]

A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 75075, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 105105
Offset: 1

Views

Author

Amiram Eldar, Dec 23 2020

Keywords

Comments

First differs from A112643, A129485 and A249263 at n = 28.

Examples

			15015 is a term since it is odd and the sum of its non-coreful divisors is A308135(15015) = 17241 > 15015.

Links

Crossrefs

Intersection of A005408 and A308127.
Cf. A308135.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A249263, A293186, A321147, A333950.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Select[Range[1, 10^5, 2], s[#] > # &]

A348605 Odd nonexponential abundant numbers: odd numbers k such that A160135(k) > k.

Original entry on oeis.org

8505, 10395, 12285, 15015, 16065, 17955, 19635, 21735, 21945, 23205, 25515, 25935, 26565, 28875, 31185, 31395, 33495, 33915, 34125, 35805, 36855, 39585, 41055, 42315, 42735, 45885, 47355, 48195, 49665, 50505, 51765, 53865, 54285, 55965, 56595, 58695, 61215, 64155
Offset: 1

Views

Author

Amiram Eldar, Oct 25 2021

Keywords

Comments

The odd terms of A348604.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 51, 360, 4117, 39803, 418663, 4099004, ... Apparently this sequence has an asymptotic density 0.0004...

Examples

			8505 is a term since A160135(8505) = 8862 > 8505.

Links

Crossrefs

Cf. A160135.
Subsequence of A005231 and A348604.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 65000, 2], q]

A379031 Odd modified exponential abundant numbers: odd numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655
Offset: 1

Views

Author

Amiram Eldar, Dec 14 2024

Keywords

Comments

First differs from its subsequences A112643 and A249263 at n = 51: a(51) = 195195 is not a term of these two sequences.
First differs from its subsequence A129485 at n = 363: a(363) = 2537535 is not a term of A129485.
First differs from A339938 at n = 28: A339938(28) = 75075 is not a term of this sequence.
First differs from A360526 at n = 46: A360526(46) = 165165 is not a term of this sequence.

Links

Crossrefs

Intersection of A005408 and A379029.
Subsequence of A005231.
Subsequences: A112643, A129485, A249263.
Cf. A241405.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526.

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[1, 10^5, 2], meAbQ]
  • PARI
    is(k) = if(!(k%2), 0, my(f=factor(k)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*k);

A380932 Odd numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

322245, 590205, 874665, 966735, 1934415, 2900205, 3224025, 3378375, 3869775, 4729725, 6081075, 6449625, 6818175, 7740495, 8783775, 8906625, 9029475, 9889425, 10135125, 10961685, 11609325, 11821425, 12900825, 13378365, 14189175, 15049125, 15481935, 15909075, 16253055
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

The odd terms in A380929.
Analogous to odd abundant numbers (A005231) with A380845 instead of A000203.

Examples

			322245 is a term since it is odd, and A380845(322245) = 679582 > 2 * 322245 = 644490.

Links

Crossrefs

Cf. A000203, A380845.
Intersection of A005408 and A380929.
Subsequence of A005231.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526, A379031.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1,10^6,2], q]
  • PARI
    isok(k) = if(!(k % 2), 0, my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k);

A127664 Infinitary amicable numbers.

Original entry on oeis.org

114, 126, 594, 846, 1140, 1260, 4320, 5940, 7920, 8460, 8640, 10744, 10856, 11760, 12285, 13500, 14595, 17700, 25728, 35712, 43632, 44772, 45888, 49308, 60858, 62100, 62700, 67095, 67158, 71145, 73962, 74784, 79296, 79650, 79750, 83142, 83904, 86400, 88730
Offset: 1

Views

Author

Ant King, Jan 26 2007

Keywords

Examples

			a(5)=1140 because 1140 is the fifth infinitary amicable number.

Links

Crossrefs

Cf. A007357, A126168, A127661, A127662, A127663, A127665, A127666, A127667, A126169, A126170, A126171.

Programs

  • Mathematica
    ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],?(And@@#&),{1}]] ]] ] Null;properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];InfinitaryAmicableNumberQ[k_]:=If[Nest[properinfinitarydivisorsum,k,2]==k && !properinfinitarydivisorsum[k]==k,True,False];Select[Range[50000],InfinitaryAmicableNumberQ[ # ] &]
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; infs[n_] := Times @@ (fun @@@ FactorInteger[n]) - n; s = {}; Do[k = infs[n]; If[k != n && infs[k] == n, AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Mar 16 2019 *)

Formula

Non-infinitary perfect numbers which satisfy A126168(A126168(n)) = n.

Extensions

More terms from Amiram Eldar, Mar 16 2019

A334975 Odd infinitary admirable numbers: the odd terms of A334974.

Original entry on oeis.org

945, 43065, 46035, 80535, 354585, 403095, 430815, 437745, 442365, 2305875, 3525795, 4404105, 4891887, 5388495, 5927985, 6126645, 6220665, 6375105, 6537375, 7853625, 8109585, 8731125, 9071865, 9338595, 9784125, 13241745, 23760555, 33381855, 34592805, 35642295
Offset: 1

Views

Author

Amiram Eldar, May 18 2020

Keywords

Comments

Of the first 10^4 infinitary admirable numbers only 9 are odd.

Crossrefs

The infinitary version of A109729.
Intersection of A005408 and A334974.
Subsequence of A127666.
Cf. A329188, A334973.

Programs

  • Mathematica
    fun[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 @@ fun @@@ FactorInteger[n]; infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e; infAdmQ[n_] := (ab = isigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && infDivQ[n, ab/2]; Select[Range[1, 5*10^5, 2], infAdmQ]

A357607 Odd numbers k such that A162296(k) > 2*k.

Original entry on oeis.org

4725, 6615, 7875, 8505, 11025, 14175, 15435, 17325, 19845, 20475, 22275, 23625, 24255, 25515, 26775, 28665, 29925, 31185, 33075, 36225, 36855, 37125, 37485, 38115, 39375, 40425, 41895, 42525, 46305, 47775, 48195, 50715, 51975, 53235, 53865, 55125, 57915, 59535
Offset: 1

Views

Author

Amiram Eldar, Oct 06 2022

Keywords

Comments

The least term that is not divisible by 3 is a(89047132) = 134785275625.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 4, 60, 640, 6650, 66044, 660230, 6604594, 66073470, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000660... .

Examples

			4725 is a term since it is odd, and A162296(4725) = 9728 > 2*4725.

Links

Crossrefs

Cf. A162296.
Subsequence of A005231, A013929 and A357605.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A348275, A348605.

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[3, 60000, 2], q]
Previous Showing 11-18 of 18 results.

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