A102443 -id:A102443 - cp's OEIS frontend

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

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

Reinhard Zumkeller, Jan 09 2005

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A102415, A102441, A102442, A102443.

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 *)

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.

A102442 Number of iterations needed to transform n by A102440 into a 3-smooth number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 2, 0, 2, 1, 1, 0, 2, 0, 2, 1, 1, 2, 3, 0, 1, 2, 0, 1, 3, 1, 3, 0, 2, 2, 1, 0, 2, 2, 2, 1, 3, 1, 3, 2, 1, 3, 4, 0, 1, 1, 2, 2, 3, 0, 2, 1, 2, 3, 4, 1, 4, 3, 1, 0, 2, 2, 3, 2, 3, 1, 4, 0, 4, 2, 1, 2, 2, 2, 3, 1, 0, 3, 4, 1, 2, 3, 3, 2, 4, 1, 2, 3, 3, 4, 2, 0, 3, 1, 2, 1, 3, 2, 3, 2, 1

Offset: 1

Reinhard Zumkeller, Jan 09 2005

nonn

			A102440(A102440(A102440(41))) = 24 and A102440(24) = 24:
41 -> 3*13 -> 3*[13->2*5] = 2*3*5 -> 2*3*[5->2*2] = 3*2^3 = 24,
therefore a(41) = 3, A102443(41) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586, A102440, A102443, A102444.

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_] := -1 + Length @ NestWhileList[f, n, FactorInteger[#][[-1, 1]] > 3 &]; Array[a, 105] (* Amiram Eldar, Feb 04 2020 after Jean-François Alcover at A102440 *)

a(n) = 0 iff n is 3-smooth (A003586);

a(A102444(n)) = n and a(m) < n for m < a(A102444(n)).

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

Reinhard Zumkeller, Jan 09 2005

See A102442, A102443 for further iterations of A102440.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A102440, A102442, A102443.

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 *)

cp's OEIS Frontend

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

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A102442 Number of iterations needed to transform n by A102440 into a 3-smooth number.

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

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