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.

A348119 Numbers k such that k and k+1 are both numbers whose powerful part is larger than their powerfree part (A328014).

Original entry on oeis.org

8, 24, 48, 49, 63, 80, 175, 224, 242, 288, 324, 350, 351, 360, 512, 539, 575, 675, 735, 832, 960, 1088, 1215, 1224, 1368, 1444, 1681, 1700, 1862, 2057, 2106, 2299, 2303, 2375, 2400, 2600, 2624, 2645, 2808, 3024, 3249, 3750, 3887, 3968, 4224, 4374, 4624, 4900, 5040
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2021

Keywords

Examples

			8 is a term since both 8 and 9 are in A328014.

Links

Crossrefs

Cf. A328014.
A060355 is a subsequence.

Programs

  • Mathematica
    f[p_, e_] := If[e==1, p, 1]; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[5040], s[#]^2 < # && s[#+1]^2 < #+1 &]

A348120 Starts of runs of 3 consecutive numbers whose powerful part is larger than their powerfree part (A328014).

Original entry on oeis.org

48, 350, 9800, 11374, 31211, 32798, 48373, 59534, 63000, 103246, 118579, 373827, 488187, 625974, 629693, 830464, 1193983, 1294298, 2989439, 3815174, 4231248, 5132699, 5331248, 6674166, 7616950, 7970157, 8388223, 8670375, 9235520, 9516680, 9841094, 11121382, 12708359
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2021

Keywords

Comments

There are no runs of 4 consecutive numbers below 10^10.
It is conjectured that there are no runs of 3 consecutive numbers that are powerful (A001694), but if they do exist, their starts are contained in this sequence.

Examples

			48 is a term since 48, 49 and 50 are all in A328014.

Links

Crossrefs

Subsequence of A328014 and A348119.
Cf. A001694.

Programs

  • Mathematica
    f[p_, e_] := If[e==1, p, 1]; s[n_] := Times @@ (f @@@ FactorInteger[n]); q[n_] := s[n]^2 < n; v = q /@ Range[3]; seq = {}; Do[v = Append[Drop[v, 1], q[k]]; If[And @@ v, AppendTo[seq, k - 2]], {k, 4, 10^6}]; seq

A328015 Decimal expansion of the growth constant for the number of terms of A328014 (numbers whose powerful part is larger than their powerfree part).

Original entry on oeis.org

1, 1, 1, 5, 4, 3, 6, 6, 3, 1, 1, 1, 1, 0, 1, 3, 6, 9, 8, 9, 3, 1, 9, 3, 4, 3, 0, 2, 9, 4, 1, 0, 9, 6, 3, 2, 7, 0, 3, 3, 2, 8, 6, 6, 4, 9, 1, 1, 3, 0, 5, 3, 1, 6, 1, 6, 7, 1, 1, 4, 7, 6, 3, 9, 5, 7, 6, 8, 0, 3, 0, 7, 0, 3, 2, 1, 1, 7, 2, 4, 6, 8, 3, 7, 7, 2, 3
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2019

Keywords

Comments

Cloutier et al. showed that the number of terms of A328014 below x is D0 * x^(3/4) + O(x^(2/3)*log(x)), where D0 is this constant.

Examples

			1.115436631111013698931934302941096327033286649113053...

Links

Crossrefs

Cf. A057521, A055231, A328014.

Programs

  • Mathematica
    $MaxExtraPrecision = 500; m = 500; f[x_] := (1 - x)*(1 + (1 - x)*x/(1 + x^(3/2))); c = LinearRecurrence[{2, -3, 2, 1, -3, 3, -1}, {0, 0, 0, -8, -5, 6, 14}, m]; RealDigits[(4/3)*(Zeta[3/2]/Zeta[3])*f[1/2]*f[1/3]*Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 3, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Formula

Equals (4/3)*(zeta(3/2)/zeta(3)) * Product_{p prime} (1 - 1/p)*(1 + (1-1/p)/(p*(1 + 1/p^(3/2)))).
Showing 1-3 of 3 results.

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

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