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 1-7 of 7 results.

A359963 Arithmetic numbers (A003601) having more divisors than all smaller arithmetic numbers.

Original entry on oeis.org

1, 3, 6, 20, 30, 60, 168, 420, 840, 1980, 2160, 2520, 7560, 10080, 15120, 27720, 79200, 83160, 110880, 166320, 262080, 332640, 554400, 786240, 831600, 1081080, 1441440, 2162160, 2882880, 4324320, 7207200, 8648640, 10810800, 17297280, 21621600, 36756720, 43243200
Offset: 1

Views

Author

Amiram Eldar, Jan 20 2023

Keywords

Comments

The corresponding numbers of divisors are 1, 2, 4, 6, 8, 12, 16, 24, 32, ... .
This sequence is infinite since there are arithmetic numbers with any number of divisors (see A359965).

Links

Crossrefs

Subsequence of A003601.
Cf. A000005, A359965.
Similar sequences: A002182, A335317, A359964.

Programs

  • Mathematica
    seq[nmax_] := Module[{s = {}, dm = 0, d}, Do[d = DivisorSigma[0, n]; If[d > dm && Divisible[DivisorSigma[1, n], d], dm = d; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^6]
  • PARI
    lista(nmax) = {my(dm = 0, d); for(n = 1, nmax, d = numdiv(n); if(d > dm && sigma(n)%d == 0, dm = d; print1(n, ", "))); }

A335316 Harmonic numbers (A001599) with a record harmonic mean of divisors.

Original entry on oeis.org

1, 6, 28, 140, 270, 672, 1638, 2970, 8190, 27846, 30240, 167400, 237510, 332640, 695520, 1421280, 2178540, 2457000, 11981970, 14303520, 17428320, 23963940, 27027000, 46683000, 56511000, 71253000, 142990848, 163390500, 164989440, 191711520, 400851360, 407386980
Offset: 1

Views

Author

Amiram Eldar, May 31 2020

Keywords

Comments

The corresponding record values are 1, 2, 3, 5, 6, 8, 9, 11, 15, ... (see the link for more values).
The terms 1, 6, 30240 and 332640 are also terms of A179971.

Examples

			The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their harmonic means of divisors (A001600) are 1, 2, 3, 5, 6, 5 and 8. The record values, 1, 2, 3, 5, 6 and 8 occur at 1, 6, 28, 140, 270 and 672, the first 6 terms of this sequence.

Links

Crossrefs

Cf. A001599, A001600, A099377, A099378, A179971, A335317, A335318.

Programs

  • Mathematica
    h[n_] := n * DivisorSigma[0, n] / DivisorSigma[1, n]; hm = 0; s = {}; Do[h1 = h[n];  If[IntegerQ[h1] && h1 > hm, hm = h1; AppendTo[s, n]], {n, 1, 10^6}]; s

A359964 Refactorable numbers (A033950) having more divisors than all smaller refactorable numbers.

Original entry on oeis.org

1, 2, 8, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3360, 5040, 10080, 15120, 20160, 25200, 30240, 55440, 100800, 110880, 221760, 277200, 443520, 665280, 720720, 1108800, 1441440, 2494800, 2882880, 3603600, 5765760, 8648640, 12972960, 14414400, 25945920, 28828800
Offset: 1

Views

Author

Amiram Eldar, Jan 20 2023

Keywords

Comments

The corresponding numbers of divisors are 1, 2, 4, 6, 8, 9, 12, 18, 20, 24, ... .
This sequence if infinite since there are refactorable numbers with arbitrarily large number of divisors. E.g., for any prime p, p^(p-1) is a refactorable number with p divisors.

Links

Crossrefs

Subsequence of A033950.
Cf. A000005, A073904, A110821.
Similar sequences: A002182, A335317, A356078, A359963.

Programs

  • Mathematica
    seq[nmax_] := Module[{s = {}, dm = 0, d}, Do[d = DivisorSigma[0, n]; If[d > dm && Divisible[n, d], dm = d; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[10^6]
  • PARI
    lista(nmax) = {my(dm = 0, d); for(n = 1, nmax, d = numdiv(n); if(d > dm && n%d == 0, dm = d; print1(n, ", "))); }

A335318 Harmonic numbers (A001599) k with a record abundancy index sigma(k)/k.

Original entry on oeis.org

1, 6, 140, 270, 672, 8190, 30240, 332640, 14303520, 17428320, 27027000, 191711520, 2144862720, 3506025600, 5943057120, 14378364000, 45578332800, 288662774400, 505159855200, 2020639420800, 10680522652800, 54557264361600
Offset: 1

Views

Author

Amiram Eldar, May 31 2020

Keywords

Comments

The corresponding record values are 1, 2, 2.4, 2.666..., 3, 3.2, 4, 4.363..., ...
The terms 1, 6, 672 and 30240 are multiply perfect numbers (A007691) with abundancy indices 1, 2, 3, and 4, respectively. There is no 5-multiperfect number (A046060) in this sequence since A046060(1) = 14182439040 is larger than the harmonic number 5943057120 which is 5-abundant, having an abundancy index 5.067...

Examples

			The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their abundancy indices are 1, 2, 2, 2.4, 2.666..., 2 and 3. The record values, 1, 2, 2.4, 2.666... and 3 occur at 1, 6, 140, 270 and 672, the first 5 terms of this sequence.

Crossrefs

Cf. A000203, A001599, A335316, A335317.

Programs

  • Mathematica
    rm = 0; s = {}; Do[h = DivisorSigma[0, n]/(r = DivisorSigma[1, n]/n); If[IntegerQ[h] && r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

A335388 Harmonic numbers (A001599) with a record number of divisors that are harmonic numbers.

Original entry on oeis.org

1, 6, 140, 672, 30240, 332640, 2178540, 17428320, 163390500, 191711520, 326781000, 1307124000, 2701389600, 8104168800, 14378364000, 40520844000, 1175104476000, 7131668544000, 49921679808000
Offset: 1

Views

Author

Amiram Eldar, Jun 04 2020

Keywords

Comments

The corresponding record values are 1, 2, 3, 4, 7, 9, 12, 15, 17, 19, 21, 25, 29, 34, 35, 43, 54, 75, 84.

Examples

			1 has one divisor which is a harmonic number: 1. 6 has 2 divisors which are harmonic numbers: 1 and 6. The next harmonic number with more than 2 divisors which are harmonic numbers is 140 with 3 such divisors: 1, 28 and 140.

Crossrefs

Cf. A001599, A335317.

Programs

  • Mathematica
    h = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; hQ[n_] := Divisible[n * DivisorSigma[0, n], DivisorSigma[1, n]]; hsigma[n_] := DivisorSum[n, Boole @ hQ[#] &]; hm = 0; s = {}; Do[h1 = hsigma[h[[k]]]; If[h1 > hm, hm = h1; AppendTo[s, h[[k]]]], {k, 1, Length[h]}]; s

A353040 Unitary harmonic numbers (A006086) with a record number of unitary divisors.

Original entry on oeis.org

1, 6, 60, 420, 5460, 791700, 93284100, 3915147600, 285805774800
Offset: 1

Views

Author

Amiram Eldar, Apr 19 2022

Keywords

Comments

The corresponding record values are 1, 4, 8, 16, 32, 64, 128, 256, 512, ...

Examples

			The first 6 unitary harmonic numbers are 1, 6, 45, 60, 90, and 420. Their numbers of unitary divisors (A034444) are 1, 4, 4, 8, 8 and 16. The record values, 1, 4, 8 and 16 occur at 1, 6, 60 and 420, the first 4 terms of this sequence.

Crossrefs

Cf. A006086, A034444, A335317.

Programs

  • Mathematica
    f[p_, e_] := 2 * p^e/(p^e + 1); r[n_] := Times @@ f @@@ FactorInteger[n]; dm = 0; s = {}; Do[h = r[n]; If[IntegerQ[h] && (d = 2^PrimeNu[n]) > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

A364726 Admirable numbers with more divisors than any smaller admirable number.

Original entry on oeis.org

12, 24, 84, 120, 672, 24384, 43065, 78975, 81081, 261261, 523776, 9124731, 13398021, 69087249, 91963648, 459818240, 39142675143, 51001180160
Offset: 1

Views

Author

Amiram Eldar, Aug 05 2023

Keywords

Comments

The corresponding numbers of divisors are 6, 8, 12, 16, 24, 28, 32, 36, 40, 48, 80, 90, 96, 120, 144, 288, 360, 480, ... .
If there are infinitely many even perfect numbers (A000396), then this sequence is infinite, because if p is a Mersenne prime exponent (A000043) and q is an odd prime that does not divide 2^p-1, then 2^(p-1)*(2^p-1)*q is an admirable number with 4*p divisors (see A165772).
a(19) > 10^11.

Crossrefs

Cf. A000005, A000043, A000396, A109745, A111592, A165772.
Similar sequences: A002182, A136404, A335008, A335317, A348198, A359963, A359964.

Programs

  • Mathematica
    admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
seq[kmax_] := Module[{s = {}, dm = 0, d1}, Do[d1 = DivisorSigma[0, k]; If[d1 > dm && admQ[k], dm = d1; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]
  • PARI
    isadm(n) = {my(ab=sigma(n)-2*n); ab>0 && ab%2 == 0 && ab/2 < n && n%(ab/2) == 0;}
lista(kmax) = {my(dm = 0, d1); for(k = 1, kmax, d1 = numdiv(k); if(d1 > dm && isadm(k), dm = d1; print1(k,", ")));}
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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