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 1-3 of 3 results.

A363189 Indices of the odd terms in the sequence of powerful numbers (A001694).

Original entry on oeis.org

1, 4, 6, 7, 10, 13, 16, 17, 20, 24, 25, 28, 30, 31, 35, 39, 41, 43, 45, 48, 51, 56, 57, 60, 62, 63, 65, 68, 71, 75, 79, 82, 83, 84, 87, 90, 94, 97, 98, 99, 102, 103, 105, 107, 110, 114, 117, 120, 122, 125, 127, 129, 133, 138, 141, 142, 144, 145, 148, 151, 152
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Comments

The asymptotic density of this sequence is (2-sqrt(2))/(3-sqrt(2)) = 0.369398... .
If A001694(k) is a term of A363190 then k and k+1 are consecutive integers in this sequence.

Examples

			The first 6 powerful numbers are 1, 4, 8, 9, 16 and 25. 1, 9 and 25 are odd and their positions in the sequence are 1, 4 and 6, respectively.

Links

Crossrefs

Cf. A001694, A062739, A363190, A363191, A363192.

Programs

  • Mathematica
    Position[Select[Range[7000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &], _?(OddQ[#] &)] // Flatten
  • PARI
    lista(kmax) = {my(c = 0); for(k = 1, kmax, if(ispowerful(k), c++; if(k%2, print1(c, ", ")))); }

A363191 a(n) is the least start of a run of exactly n consecutive powerful numbers (A001694) that are even, or -1 if no such run exists.

Original entry on oeis.org

16, 4, 196, 968, 8712, 437400, 85730400, 5030690600, 264615012500, 5239012864, 550886816376, 2494017320776852
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Comments

No more terms below 10^18.
At most one of the n even consecutive powerful numbers in the run is a perfect square. - David A. Corneth, May 21 2023

Examples

			a(1) = 16, since 16 = 2^4 is an even powerful number, preceded by an odd powerful number, 9 = 3^2, and followed by an odd powerful number, 25 = 5^2.
a(2) = 4, since 4 = 2^2 and 8 = 2^3 are two consecutive even powerful numbers, preceded by an odd powerful number, 1, and followed by an odd powerful number, 9 = 3^2.

Crossrefs

Cf. A001694, A062739, A349062, A363189, A363190, A363192.

Programs

  • Mathematica
    seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, lim^(1/3)}, {i, 1, Sqrt[lim/j^3]}]]], s = {}, rem, ind}, rem = Mod[pow, 2]; Do[ind = SequencePosition[rem, Join[{1}, Table[0, {k}], {1}], 1]; If[ind == {}, Break[]]; AppendTo[s, pow[[ind[[1, 1]] + 1]]], {k, 1, Infinity}]; s]; seq[10^10]

A363192 a(n) is the least start of a run of exactly n consecutive powerful numbers (A001694) that are odd, or -1 if no such run exists.

Original entry on oeis.org

1, 25, 2187, 703125, 93096125, 10229709861, 197584409639, 32044275110699, 164029657560618375
Offset: 1

Views

Author

Amiram Eldar, May 21 2023

Keywords

Comments

No more terms below 10^18.
At most one of the n odd consecutive powerful numbers in the run is a perfect square. - David A. Corneth, May 21 2023

Examples

			a(1) = 1, since 1 is an odd powerful number, followed by an even powerful number, 4 = 2^2.
a(2) = 25, since 25 = 5^2 and 27 = 3^3 are two consecutive odd powerful numbers, preceded by an even powerful number, 16 = 2^4, and followed by an even powerful number, 32 = 2^5.

Crossrefs

Cf. A001694, A062739, A349062, A363189, A363190, A363191.

Programs

  • Mathematica
    seq[lim_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, lim^(1/3)}, {i, 1, Sqrt[lim/j^3]}]]], s = {}, rem, ind}, rem = Join[{0}, Mod[pow, 2]]; Do[ind = SequencePosition[rem, Join[{0}, Table[1, {k}], {0}], 1]; If[ind == {}, Break[]]; AppendTo[s, pow[[ind[[1, 1]]]]], {k, 1, Infinity}]; s]; seq[1.1*10^10]
Showing 1-3 of 3 results.

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

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