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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A176802 a(n) = the smallest natural numbers m such that product of harmonic mean of the divisors of n and harmonic mean of the divisors of m are integers.

Original entry on oeis.org

1, 3, 2, 7, 28, 1, 4, 420, 182, 27, 270, 14, 126, 4, 6, 31, 1638, 91, 980, 7, 32, 84, 30240, 15, 248, 63, 10, 1, 8190, 3, 16, 21, 672, 819, 4, 60515, 117800, 420, 840, 84, 55860, 4, 332640, 42, 182, 1638, 30240, 62, 380, 744, 270, 4655, 167400, 5, 54, 60, 980
Offset: 1

Views

Author

Jaroslav Krizek, Apr 26 2010

Keywords

Comments

Harmonic mean of the divisors of number n is rational number b(n) = n*A000005(n) / A000203(n) = A099377(n) / A099378(n).
a(n) = 1 for infinitely many n. a(n) = 1 for numbers from A001599: a(A001599(n)) = 1. a(n) = 1 iff A099378(n) = 1.

Examples

			For n = 4; b(4) = 12/7, a(4) = 7 because b(7) = 7/4; 12/7 * 7/4 = 3 (integer).

Links

Crossrefs

Cf. A000005, A000203, A001599, A099377, A099378.

Programs

  • Mathematica
    h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; a[n_] := Module[{hn = h[n], k = 1}, While[! IntegerQ[hn * h[k]], k++]; k]; Array[a, 35] (* Amiram Eldar, Mar 22 2024 *)
  • PARI
    h(n) = {my(f = factor(n)); numdiv(f)/sigma(f, -1);}
a(n) = {my(hn = h(n), k = 1); while(denominator(hn * h(k)) > 1, k++); k;} \\ Amiram Eldar, Mar 22 2024

Extensions

Data corrected and extended by Amiram Eldar, Mar 22 2024

A335291 Numbers m such that the delta(m) = abs(h(m+1) - h(m)) is smaller than delta(k) for all k < m, where h(m) is the harmonic mean of the divisors of m.

Original entry on oeis.org

1, 2, 4, 91, 272, 20118, 20712, 33998, 42818, 61695, 25274946, 27194929, 34883654, 40406622, 43176318, 47350866, 52680050, 149736013, 154957034, 162929406, 171560153, 187012577, 208015843, 267361097, 300087726, 325189758, 355153181, 443360633, 584803578, 605883413
Offset: 1

Views

Author

Amiram Eldar, May 30 2020

Keywords

Comments

Apparently, most of the terms m have h(m+1) > h(m) and numerator(delta(m)) = 1.
Can two consecutive numbers have the same harmonic mean of divisors? If yes, then this sequence is finite.

Examples

			The values of delta(k) for the first terms are 0.333..., 0.166..., 0.047..., 0.0357..., ...

Links

Crossrefs

Cf. A000005, A000203, A099377, A099378.
Cf. A238380, A335071.

Programs

  • Mathematica
    h[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; dm = 1; h1 = h[1]; s = {}; Do[h2 = h[n]; d = Abs[h2 - h1]; If[d < dm, dm = d; AppendTo[s, n-1]]; h1 = h2, {n, 2, 10^5}]; s

A346400 Composite numbers k such that the numerator of the harmonic mean of the divisors of k is equal to k.

Original entry on oeis.org

20, 21, 22, 27, 35, 38, 39, 45, 49, 55, 56, 57, 65, 68, 77, 85, 86, 93, 99, 110, 111, 115, 116, 118, 119, 125, 129, 133, 134, 143, 147, 150, 155, 161, 164, 166, 169, 183, 184, 185, 187, 189, 201, 203, 205, 207, 209, 212, 214, 215, 217, 219, 221, 235, 237, 245
Offset: 1

Views

Author

Amiram Eldar, Nov 01 2021

Keywords

Comments

Composite numbers k such that A099377(k) = k.
Since the harmonic mean of the divisors of an odd prime p is p/((p+1)/2), its numerator is equal to p. Therefore, this sequence is restricted to composite numbers.
This sequence is infinite. For example, if p is a prime of the form 8*k+3 (A007520) with k>1, then 2*p is a term.

Examples

			20 is a term since the harmonic mean of the divisors of 20 is 20/7.

Links

Crossrefs

Intersection of A002808 and A250094.
Cf. A099377, A099378.

Programs

  • Mathematica
    q[n_] := CompositeQ[n] && Numerator[DivisorSigma[0, n]/DivisorSigma[-1, n]] == n; Select[Range[250], q]
  • PARI
    isok(k) = my(d=divisors(k)); (#d>2) && (numerator(#d/sum(i=1, #d, 1/d[i])) == k); \\ Michel Marcus, Nov 01 2021
  • PARI
    list(lim)=my(v=List()); forfactored(n=20,lim\1, if(vecsum(n[2][,2])>1 && numerator(sigma(n,0)/sigma(n,-1))==n[1], listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 01 2021

A348828 Numbers that are equal to the product of the numerator and denominator of the harmonic mean of their divisors.

Original entry on oeis.org

1, 30, 138, 210, 2280, 4676, 5970, 6972, 8372, 10290, 12012, 12306, 20370, 22386, 105420, 116844, 118524, 153480, 189420, 195860, 204204, 218430, 289560, 293880, 362180, 369740, 408510, 414990, 494760, 525420, 629640, 933660, 952770, 1529010, 1564332, 1647810
Offset: 1

Views

Author

Amiram Eldar, Nov 01 2021

Keywords

Comments

Numbers k such that A099377(k) * A099378(k) = k.
Is 1 the only odd term? There are no other odd terms below 3*10^9.

Examples

			30 is a term since the harmonic mean of its divisors is 10/3 and 10*3 = 30.
138 is a term since the harmonic mean of its divisors is 23/6 and 23*6 = 138.

Links

Crossrefs

Cf. A099377, A099378.

Programs

  • Mathematica
    q[n_] := Numerator[(hm = DivisorSigma[0, n]/DivisorSigma[-1, n])] * Denominator[hm] == n; Select[Range[10^6], q]
  • PARI
    isok(k) = my(d=divisors(k), h=#d/sum(i=1, #d, 1/d[i])); k == numerator(h)*denominator(h); \\ Michel Marcus, Nov 01 2021

A349478 a(n) is the least number k such that the sequence of elements of the continued fraction of the harmonic mean of the divisors of k is palindromic with length n, or -1 if no such k exists.

Original entry on oeis.org

1, 15, 8, 545, 21, 1131, 16, 98124, 28676, 1109305, 28672, 16837500, 1231932, 477021580, 6129711, 734420331, 441972042, 4343866215, 42741916965, 96692841558, 2193739177
Offset: 1

Views

Author

Amiram Eldar, Nov 19 2021

Keywords

Comments

a(23) = 60755428490.
No more terms below 10^11.

Examples

			The elements of the continued fractions of the harmonic mean of the divisors of the terms are:
   n         a(n)   elements
  --  -----------   -------------------------------------------
   1            1   1
   2           15   2,2
   3            8   2,7,2
   4          545   3,3,3,3
   5           21   2,1,1,1,2
   6         1131   5,2,1,1,2,5
   7           16   2,1,1,2,1,1,2
   8        98124   17,1,1,3,3,1,1,17
   9        28676   6,1,2,3,1,3,2,1,6
  10      1109305   6,1,1,1,1,1,1,1,1,6
  11        28672   11,2,1,1,1,10,1,1,1,2,11
  12     16837500   24,1,1,1,2,1,1,2,1,1,1,24
  13      1231932   18,1,1,1,1,1,8,1,1,1,1,1,18
  14    477021580   38,2,3,1,1,1,1,1,1,1,1,3,2,38
  15      6129711   14,2,2,1,1,1,1,9,1,1,1,1,2,2,14
  16    734420331   20,2,1,1,1,1,1,1,1,1,1,1,1,1,2,20
  17    441972042   15,1,3,2,2,1,1,2,15,2,1,1,2,2,3,1,15
  18   4343866215   18,1,1,7,1,8,2,1,1,1,1,2,8,1,7,1,1,18
  19  42741916965   94,1,1,7,4,1,1,1,1,3,1,1,1,1,4,7,1,1,94
  20  96692841558   28,2,4,1,1,4,1,1,1,6,6,1,1,1,4,1,1,4,2,28
  21   2193739177   19,1,1,1,3,1,1,1,1,1,9,1,1,1,1,1,3,1,1,1,19

Crossrefs

Cf. A099377, A099378, A349473, A349477.

Programs

  • Mathematica
    cfhm[n_] := ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i, cf}, While[c < len && n < nmax, cf = cfhm[n]; If[PalindromeQ[cf] && (i = Length[cf]) <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[11, 10^7]

A349498 a(n) is the least number k such that A349497(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 6, 24, 170, 140, 270, 1140, 630, 1400, 4420, 2016, 8680, 11704, 18620, 8190, 20196, 12960, 90860, 13860, 30800, 55860, 148770, 51408, 30240, 78120, 242060, 153120, 282555, 65520, 564564, 268128, 381150, 798560, 592515, 535680, 1503216, 318240, 664020, 726180, 790020
Offset: 1

Views

Author

Amiram Eldar, Nov 20 2021

Keywords

Examples

			The elements of the continued fractions of the harmonic mean of the divisors of the first 10 terms are:
   n  a(n)  elements
  --  ----  --------
   1     1  1
   2     6  2
   3    24  3,5
   4   170  4,5,16
   5   140  5
   6   270  6
   7  1140  8,7
   8   630  8,13
   9  1400  9,31
  10  4420  10,44,10

Crossrefs

Cf. A099377, A099378, A349473, A349497.

Programs

  • Mathematica
    f[n_] := Min[ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[25, 10^6]

A377357 Numbers k with the property that the smallest subpart of the symmetric representation of sigma(k) equals the denominator of the harmonic means of the divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 16, 17, 19, 23, 26, 28, 29, 30, 31, 34, 37, 41, 43, 47, 52, 53, 58, 59, 61, 64, 67, 71, 73, 74
Offset: 1

Views

Author

Omar E. Pol, Oct 26 2024

Keywords

Comments

Numbers k such that A296513(k) = A099378(k).
For the definition of the "subpart" see A279387.
For a diagram with the subparts for the first 16 positive integers see A296508.

Crossrefs

Cf. A000396, A008578, A099378, A196020, A235791, A236104, A237270, A237591, A237593, A279387, A296508.
Previous Showing 41-47 of 47 results.

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