A071830 Smallest k > n such that Lpf(n) = Lpf(k) where Lpf(x) denotes the largest prime factor in x factorization.

4, 6, 8, 10, 9, 14, 16, 12, 15, 22, 18, 26, 21, 20, 32, 34, 24, 38, 25, 28, 33, 46, 27, 30, 39, 36, 35, 58, 40, 62, 64, 44, 51, 42, 48, 74, 57, 52, 45, 82, 49, 86, 55, 50, 69, 94, 54, 56, 60, 68, 65, 106, 72, 66, 63, 76, 87, 118, 75, 122, 93, 70, 128, 78, 77, 134, 85, 92

Offset

2,1

Comments

From Rémy Sigrist, Jun 03 2017: (Start)

This sequence is a permutation of the composite numbers (A002808).

a(p) = 2*p for any prime p.

a(2^k) = 2^(k+1) for any n > 0.

For any prime p and n >= 0, a^n(p)/p is the (n+1)-th p-smooth number (where a^n denotes the n-th iterate of a).

a(n) <= 2*n for any n > 1 (as Lpf(2*n) = Lpf(n)).

See also A287932 for the least prime factor equivalent.

(End)

Links

Rémy Sigrist, Table of n, a(n) for n = 2..10000

Formula

a(n) = A071829(n) + n. - Sean A. Irvine, Aug 16 2024

Mathematica

Array[Which[PrimeQ[#], 2 #, IntegerQ@ Log2[#], 2^(IntegerExponent[#, 2] + 1), True, If[#1 <= #2^2, (#1/#2 + 1) #2, Block[{k = #1/#2 + 1}, While[FactorInteger[k][[-1, 1]] > #2, k++]; k #2]] & @@ {#, FactorInteger[#][[-1, 1]]}] &[#] &, 68, 2] (* Michael De Vlieger, Nov 03 2021 *)
Lpf[x_]:=FactorInteger[x][[-1, 1]]; Array[(k=#; While[Lpf@#!=Lpf@++k]; k)&, 68, 2] (* Giorgos Kalogeropoulos, Nov 03 2021 *)

Prog

(PARI) for(n=2, 120, s=+1; while(abs(component(component(factor(n), 1), omega(n))-component(component(factor(n+s), 1), omega(n+s)))>0, s++); print1(n+s, ", "))
(PARI) a(n) = { my(f = factor(n)[, 1], h = f[#f], s = n\h); for(i = s+1, oo, c = factor(i)[, 1]; if(c[#c] <= h, return(i*h) ) ) } \\ David A. Corneth, Nov 03 2021

Crossrefs

Cf. A002808, A006530, A071829, A287932.

Sequence in context: A161396 A145256 A087789 * A276982 A340846 A167146

Adjacent sequences: A071827 A071828 A071829 * A071831 A071832 A071833

Keyword

nonn,easy

Author

Benoit Cloitre, Jun 08 2002

Status

approved