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 1-4 of 4 results.

A102643 A006530(x)=2 is a local minimum if x=2^n. Running upward with argument x, the largest prime divisor should increase. The value of first peak is a(n).

Original entry on oeis.org

3, 5, 11, 17, 17, 13, 43, 257, 257, 41, 683, 4099, 2731, 2731, 331, 65537, 65537, 262147, 174763, 174763, 61681, 199729, 2796203, 2796203, 4051, 9586981, 87211, 15790321, 15790321, 1073741827, 715827883, 715827883, 6700417, 26317, 86171
Offset: 1

Views

Author

Labos Elemer, Jan 21 2005

Keywords

Comments

We may call these terms "upward-zenith-primes" belonging to 2^n-s. They do not exceed next-primes after 2^n [A014210(n)].

Examples

			n=22: 2^22=4194304; largest prime divisors for n+j, j=0, 1, 2, ... are {2, 2113, 5419, 16981, 61681, 199729, 7109}. The first peak after 2^22=4194304 is a(22)=199729.

Links

Crossrefs

Cf. A006530, A102640, A102641, A102642, A102644, A014210.

Programs

  • Mathematica
    Table[2 + Total@ TakeWhile[Differences@ Array[FactorInteger[#][[-1, 1]] &, 20, 2^n], # > 0 &], {n, 35}] (* Michael De Vlieger, Jul 31 2017 *)

A102640 Compute the greatest prime factors (GPFs, A006530()) of j + 2^n for j = 0, 1, ..., L. a(n) is the maximal length L of such a sequence in which the greatest prime factors are increasing with increasing j.

Original entry on oeis.org

2, 2, 4, 2, 3, 2, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 3, 4, 2, 3, 3, 6, 2, 3, 2, 4, 2, 2, 3, 4, 2, 3, 3, 2, 2, 4, 2, 4, 2, 2, 2, 4, 2, 3, 4, 4, 2, 4, 2, 3, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 4, 4, 2, 4, 2, 3, 2, 2, 4, 4, 2, 3, 3, 4, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 3, 2, 2, 2, 3, 4, 2, 4, 2, 3, 2, 2, 3
Offset: 1

Views

Author

Labos Elemer, Jan 21 2005

Keywords

Comments

A006530(2^n)=2 is a local minimum. Going either upward or downward with the argument, the greatest prime factors are increasing for a while.

Examples

			For n = 12: 2^10 = 4096. The greatest prime factors of 4096, 4097, 4098, 4099 are as follows: {2, 241, 683, 4099}. A006530(4100) = 41 is already smaller than A006530(4099). Thus the length of increasing GPF sequence is 4 = a(12).

Crossrefs

Cf. A006530, A102641, A102642, A102643, A102644.

Programs

  • Mathematica
    With[{nn = 12}, Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Array[FactorInteger[#][[-1, 1]] &, nn, k]][2^n], {n, 105}]] (* Michael De Vlieger, Jul 24 2017 *)

A102641 Compute the greatest prime factors (GPFs, A006530()) of -j + 2^n for j = 0, 1, ..., L. a(n) is the maximal length L of such a sequence in which the greatest prime factors are increasing with decreasing j.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 3, 2, 4, 4, 4, 2, 3, 4, 4, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 6, 2, 4, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 6, 4, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 4, 4, 4, 2, 3, 3, 4, 2, 4, 2, 3, 5
Offset: 1

Views

Author

Labos Elemer, Jan 21 2005

Keywords

Comments

A006530(2^n)=2 is a local minimum. Going either upward or downward with the argument, the largest prime factors are increasing for a while. Here the maximal length of increasing greatest-prime-factor sequences are given when going downward with the arguments. Compare with A102640.

Examples

			For n = 12: 2^10 = 4096. The greatest prime factors of 4096, 4095, 4094, 4093 are as follows: {2, 13, 89, 4093}. A006530(4092) = 31 is already smaller than A006530(4093). Thus the length of the increasing GPF sequence is 4 = a(12).

Crossrefs

Cf. A006530, A102640, A102642, A102643, A102644.

Programs

  • Mathematica
    With[{nn = 12}, Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Table[FactorInteger[m][[-1, 1]], {m, k, k - nn, -1}]][2^n], {n, 105}] ] (* Michael De Vlieger, Jul 24 2017 *)

A102642 a(n) = A102640(n) + A102641(n) - 1.

Original entry on oeis.org

2, 3, 5, 5, 4, 5, 3, 4, 4, 5, 3, 7, 3, 6, 3, 3, 4, 6, 3, 6, 4, 7, 3, 6, 3, 5, 3, 5, 4, 7, 3, 5, 4, 5, 5, 7, 3, 6, 5, 5, 3, 6, 3, 6, 5, 5, 3, 7, 3, 5, 4, 5, 3, 6, 7, 3, 6, 3, 3, 4, 3, 5, 4, 3, 4, 6, 3, 5, 5, 6, 3, 7, 3, 5, 7, 5, 5, 6, 3, 5, 6, 5, 3, 4, 3, 5, 3, 6, 3, 5, 3, 5, 4, 5, 5, 5, 4, 6, 4, 7, 3, 6, 3, 4, 7
Offset: 1

Views

Author

Labos Elemer, Jan 21 2005

Keywords

Comments

A006530(2^n)=2 is a local minimum. Actual sequence displays the "width of valley" between the two nearest peaks of largest prime divisors. At the bottom of valley lies the number 2, the minimum.

Examples

			n=12: 2^10=4096. The greatest prime divisors of numbers around 4096 [both downward and upward] are as follows: {31, 4093, 89, 13, 2, 241, 683, 4099, 41}. The length of relevant sequence, i.e., between peaks 4093 and 4099 is 7, thus a(12)=7.

Crossrefs

Cf. A006530, A102640, A102641, A102643, A102644.

Programs

  • Mathematica
    With[{nn = 12, lim = 105}, Map[Total@ # - 1 &, Transpose@ {Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Array[FactorInteger[#][[-1, 1]] &, nn, k]][2^n], {n, lim}], Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Table[FactorInteger[m][[-1, 1]], {m, k, k - nn, -1}]][2^n], {n, lim}]}]] (* Michael De Vlieger, Jul 30 2017 *)
Showing 1-4 of 4 results.

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