A102440 -id:A102440 - cp's OEIS frontend

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

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)).

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

Reinhard Zumkeller, Jan 09 2005

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

			See A102442.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A003586, A102440, A102442.

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 *)

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

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 *)

A102415 Greatest semiprime less than n-th prime.

Original entry on oeis.org

4, 6, 10, 10, 15, 15, 22, 26, 26, 35, 39, 39, 46, 51, 58, 58, 65, 69, 69, 77, 82, 87, 95, 95, 95, 106, 106, 111, 123, 129, 134, 134, 146, 146, 155, 161, 166, 169, 178, 178, 187, 187, 194, 194, 209, 221, 226, 226, 226, 237, 237, 249, 254, 262, 267, 267, 274, 278, 278

Offset: 3

Reinhard Zumkeller, Jan 08 2005

nonn

			a(3) = 4 since 4 is the greatest semiprime less than prime(3) = 5.

Amiram Eldar, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A102414, A102440.

a[n_] := Module[{m = Prime[n] - 1}, While[PrimeOmega[m] != 2, m--]; m]; Array[a, 60, 3] (* Amiram Eldar, Feb 06 2020 *)

a(n) = {sp = prime(n)-1; while(bigomega(sp) != 2, sp--); sp;} \\ Michel Marcus, Mar 04 2017

a(n) < A000040(n) < A102414(n).

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

Reinhard Zumkeller, Jan 09 2005

nonn

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

Cf. A102440.

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

More terms from David Wasserman, Apr 04 2008

cp's OEIS Frontend

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

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

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

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A102415 Greatest semiprime less than n-th prime.

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

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