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.

A102443 a(n)=b(n, A102442(n)), where b(n,0)=n and b(n,k+1)=A102440(b(n,k)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 8, 12, 8, 12, 12, 16, 12, 18, 12, 16, 18, 16, 16, 24, 16, 16, 27, 24, 16, 24, 16, 32, 24, 24, 24, 36, 24, 24, 24, 32, 24, 36, 24, 32, 36, 32, 32, 48, 36, 32, 36, 32, 36, 54, 32, 48, 36, 32, 32, 48, 32, 32, 54, 64, 32, 48, 32, 48, 48, 48, 48, 72, 48
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 09 2005

Keywords

Comments

a(a(n)) = A102440(a(n)) = a(n).
Completely multiplicative because A102440 is. The conversion of every prime into a 3-smooth number is independent of any other prime. - Andrew Howroyd, Jul 31 2018

Examples

			See A102442.

Links

Crossrefs

Cf. A003586, A102440, A102442.

Programs

  • Mathematica
    g[p_] := (* greatest semiprime less than prime p *) g[p] = For[k = p - 1, True, k--, If[PrimeOmega[k] == 2, Return[k]]];
A102440[n_] := Product[{p, e} = pe; If[p <= 3, p, g[p]]^e, {pe, FactorInteger[n]}];
A102442[n_] := Length[NestWhileList[A102440, n, FactorInteger[#][[-1, 1]] > 3 & ] - 1];
a[n_] := b[n, A102442[n]];
b[n_, 0] := n;
b[n_, k_] := A102440[b[n, k - 1]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 16 2021 *)
  • PARI
    a(n)={while(1, my(f=factor(n)); if(!#select(t->t>3, f[,1]), return(n), n=prod(i=1, #f~, my(p=f[i,1]); while(p>4 && bigomega(p)<>2, p--); p^f[i,2])))} \\ Andrew Howroyd, Jul 31 2018

A102444 Smallest number m such that A102442(m)=n.

Original entry on oeis.org

1, 5, 11, 23, 47, 149, 359, 719, 1439, 2879, 12097, 24197, 48407, 96821, 193649, 968237, 2614327, 5809201, 11618413, 25055857, 75167579, 225502703, 451005407, 2109888899, 4510054327, 14500925539, 43502776619, 87005553241, 174011106487
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 09 2005

Keywords

Comments

Primes: a(n+1) > 2*a(n).

Crossrefs

Cf. A102440.
A056637 describes another iteration that reduces primes to 2 and 3.

Extensions

More terms from David Wasserman, Apr 04 2008

A102440 Replace each prime factor of n that is greater than 3 with the greatest semiprime less than it.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 10, 12, 10, 12, 12, 16, 15, 18, 15, 16, 18, 20, 22, 24, 16, 20, 27, 24, 26, 24, 26, 32, 30, 30, 24, 36, 35, 30, 30, 32, 39, 36, 39, 40, 36, 44, 46, 48, 36, 32, 45, 40, 51, 54, 40, 48, 45, 52, 58, 48, 58, 52, 54, 64, 40, 60, 65, 60, 66, 48, 69, 72, 69
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 09 2005

Keywords

Examples

			a(99) = a(3*3*11) -> 3*3*[11->2*5] = 3*3*2*5 = 90.

Links

Crossrefs

Cf. A102415, A102441, A102442, A102443.

Programs

  • Mathematica
    g[p_] := (* greatest semiprime less than prime p *) g[p] = For[k = p - 1, True, k--, If[PrimeOmega[k] == 2, Return[k]]];
a[n_] := Product[{p, e} = pe; If[p <= 3, p, g[p]]^e, {pe, FactorInteger[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 27 2019 *)

Formula

Multiplicative with prime(i) -> (if i<=2 then prime(i) else A102415(i)).
a(n) <= n and a(n) = n iff n is 3-smooth, see A003586.
A102441(n) = a(a(n)), see A102442, A102443 for iterations.

A102441 a(n) = A102440(A102440(n)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 6, 8, 9, 8, 8, 12, 8, 12, 12, 16, 12, 18, 12, 16, 18, 16, 20, 24, 16, 16, 27, 24, 20, 24, 20, 32, 24, 24, 24, 36, 24, 24, 24, 32, 30, 36, 30, 32, 36, 40, 44, 48, 36, 32, 36, 32, 45, 54, 32, 48, 36, 40, 52, 48, 52, 40, 54, 64, 32, 48, 40, 48, 60, 48, 66, 72, 66
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 09 2005

Keywords

Comments

See A102442, A102443 for further iterations of A102440.

Links

Crossrefs

Cf. A102440, A102442, A102443.

Programs

  • Mathematica
    g[p_] := g[p] = For[k = p - 1, True, k--, If[PrimeOmega[k] == 2, Return[k]]]; f[n_] := Product[{p, e} = pe; If[p <= 3, p, g[p]]^e, {pe, FactorInteger[n]}]; a[n_] := f[f[n]]; Array[a, 100] (* Amiram Eldar, Feb 04 2020 after Jean-François Alcover at A102440 *)
Showing 1-4 of 4 results.

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