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 1-5 of 5 results.

A349499 Numbers k such that A349474(k) = A349474(k+1).

Original entry on oeis.org

2, 7, 10, 14, 19, 22, 56, 59, 60, 63, 65, 66, 67, 68, 69, 70, 76, 77, 81, 104, 109, 113, 114, 123, 125, 137, 138, 154, 155, 164, 171, 184, 185, 187, 190, 195, 199, 210, 217, 221, 230, 232, 236, 248, 251, 255, 257, 274, 276, 280, 281, 282, 290, 293, 295, 301, 306
Offset: 1

Views

Author

Amiram Eldar, Nov 20 2021

Keywords

Examples

			2 is a term since A349474(2) = A349474(3) = 2.
7 is a term since A349474(7) = A349474(8) = 3.

Links

Crossrefs

Cf. A349473, A349474.

Programs

  • Mathematica
    c[n_] := Length @ ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]; Select[Range[350], c[#] == c[# + 1] &]

A349500 a(n) is the least number k such that A349474(k) = A349474(k+1) = n, or -1 if no such k exists.

Original entry on oeis.org

2, 7, 76, 56, 81, 63, 913, 892, 1969, 4824, 22855, 16819, 48922, 170649, 273216, 607783, 1204354, 1910608, 3433671, 10104969, 19546522, 21424744, 66961728, 103366113, 217458328, 813832568, 771821712, 2370545332, 4638470426, 7190276806, 9309810824, 35730615937
Offset: 2

Views

Author

Amiram Eldar, Nov 20 2021

Keywords

Comments

The sequence begins at n = 2. a(1) != -1 if and only if two consecutive harmonic numbers exist. There are no odd harmonic numbers between 2 and 10^24 (Cohen and Sorli, 2010) and it is conjectured that they do not exist.

Examples

			a(2) = 2 since A349474(2) = A349474(3) = 2 and there is no smaller pair of consecutive numbers with this property.
a(3) = 7 since A349474(7) = A349474(8) = 3 and there is no smaller pair of consecutive numbers with this property.

Links

Crossrefs

Cf. A001599, A349474, A349499, A349501.

Programs

  • Mathematica
    c[n_] := Length @ ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], k = 1, n = 1, i}, s[[1]] = -1; While[n < nmax && k < len, i = c[n]; If[c[n+1] == i && i <= len && s[[i]] == 0, k++; s[[i]] = n]; n++]; Rest @ s]; seq[15, 10^6]

A349501 a(n) is the least start of a run of exactly n consecutive numbers with the same length of the continued fraction of the harmonic mean of their divisors (A349474).

Original entry on oeis.org

1, 2, 59, 280, 3539, 57575, 65, 15410548, 9286977451, 24510585369
Offset: 1

Views

Author

Amiram Eldar, Nov 20 2021

Keywords

Examples

			a(2) = 2 since A349474(2) = A349474(3) = 2 and there is no smaller pair of consecutive numbers with this property.
a(3) = 59 since A349474(59) = A349474(60) = A349474(61) = 3 and there is no smaller triple of consecutive numbers with this property.

Crossrefs

Cf. A349474, A349499, A349500.

Programs

  • Mathematica
    d[n_] := Length @ ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; seq[len_, nmax_] := Module[{s = Table[0, {len}], dprev = -1, n = 1, c = 0, k = 0}, While[k < len && n < nmax, d1 = d[n]; If[d1 == dprev, c++, If[c > 0 && c <= len && s[[c]] == 0, k++; s[[c]] = n - c]; c = 1]; n++; dprev = d1]; TakeWhile[s, # > 0 &]]; seq[7, 10^5]

A349475 a(n) is the least number k such that A349474(k) = n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 5, 4, 21, 25, 16, 36, 106, 712, 1588, 3775, 900, 4356, 18496, 14400, 45700, 87003, 135445, 229543, 554216, 937019, 1764724, 3431952, 3431088, 10217808, 21357233, 36972202, 42436276, 79056144, 235027304, 261540000, 530582544, 705929608, 1371526825, 1127941321
Offset: 1

Views

Author

Amiram Eldar, Nov 19 2021

Keywords

Examples

			a(3) = 5 since 5 is the least number k such that A349474(k) = 3.

Crossrefs

Cf. A071865, A349473, A349474.

Programs

  • Mathematica
    cflen[n_] := Length @ 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 = cflen[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; TakeWhile[s, # > 0 &]]; seq[20, 10^7]

A349473 Irregular triangle read by rows: the n-th row contains the elements in the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 2, 7, 2, 2, 13, 2, 4, 2, 1, 1, 5, 2, 1, 1, 3, 1, 1, 6, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 8, 2, 1, 3, 3, 1, 1, 9, 2, 1, 6, 2, 1, 1, 1, 2, 2, 2, 4, 1, 1, 11, 3, 5, 2, 2, 2, 1, 1, 2, 2, 2, 10, 2, 1, 2, 3, 3, 1, 1, 14
Offset: 1

Views

Author

Amiram Eldar, Nov 19 2021

Keywords

Comments

For an odd prime p > 3, the p-th row has a length 3 with a(p, 1) = a(p, 2) = 1 and a(p, 3) = (p-1)/2.
For a harmonic number m = A001599(k), the m-th row has a length 1 with a(k, 1) = A099377(m) = A001600(k).

Examples

			The first ten rows of the triangle are:
  1,
  1, 3,
  1, 2,
  1, 1, 2, 2,
  1, 1, 2,
  2,
  1, 1, 3,
  2, 7, 2,
  2, 13,
  2, 4, 2
  ...

Crossrefs

Cf. A001599, A001600, A099377, A099378.
Cf. A349474 (row lengths).

Programs

  • Mathematica
    row[n_] := ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; Table[row[k], {k, 1, 29}] // Flatten
Showing 1-5 of 5 results.

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

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