|
%I #15 Jun 07 2024 10:40:33
%S 4,9,8,25,8,49,16,27,16,121,16,169,16,25,32,289,24,361,25,27,32,529,
%T 27,125,32,81,32,841,32,961,64,81,64,49,48,1369,64,81,50,1681,48,1849,
%U 64,75,64,2209,54,343,64,81,64,2809,64,121,64,81,64,3481,64,3721
%N a(n) = least integer k > n such that any prime factor of k is also a prime factor of n.
%C In other words:
%C - a(n) is the least k > n such that rad(k) divides rad(n), where rad = A007947,
%C - or, if P_n denotes the set of prime factors of n, then a(n) is the least P_n-smooth number > n.
%C For any n > 1, n < a(n) <= n*lpf(n), where lpf = A020639.
%C a(p^k) = p^(k+1) for any prime p and k > 0.
%C a(n) is never squarefree.
%C This sequence has connections with A079277:
%C - here we search the least P_n-smooth number > n, there the largest < n,
%C - also, if omega(n) > 1 (where omega = A001221),
%C then n/lpf(n) < A001221(n) < n,
%C so n < A001221(n)*lpf(n) < n*lpf(n),
%C as A001221(n)*lpf(n) is P_n-smooth,
%C we have a(n) <= A001221(n)*lpf(n) < n*lpf(n),
%C and n cannot divide a(n).
%C The (logarithmic) scatterplot of the sequence has horizontal rays similar to those observed for A079277; they correspond to frequent values, typically with a small number of distinct prime divisors (see also scatterplots in Links section).
%C Given n < a(n) <= n*lpf(n), a(n) | n^m with m >= 2. Values of m: {2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 3, 5, 2, 3, 2, ...}. - _Michael De Vlieger_, Jul 02 2017
%H Rémy Sigrist, <a href="/A289280/b289280.txt">Table of n, a(n) for n = 2..10000</a>
%H Rémy Sigrist, <a href="/A289280/a289280.gp.txt">PARI program for A289280</a>
%H Rémy Sigrist, <a href="/A289280/a289280.png">Scatterplot of the ordinal transform of the first 100000 terms</a>
%H Rémy Sigrist, <a href="/A289280/a289280_1.png">Logarithmic scatterplot of the first 100000 terms</a>
%e For n = 42:
%e - 42 = 2 * 3 * 7, hence P_42 = { 2, 3, 7 },
%e - the P_42-smooth numbers are: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 27, 28, 32, 36, 42, 48, 49, ...
%e - hence a(42) = 48.
%e From _Michael De Vlieger_, Jul 02 2017: (Start)
%e a(n) divides n^m with m >= 2:
%e n a(n) m
%e 2 4 2
%e 3 9 2
%e 4 8 2
%e 5 25 2
%e 6 8 3
%e 7 49 2
%e 8 16 2
%e 9 27 2
%e 10 16 4
%e 11 121 2
%e 12 16 2
%e 13 169 2
%e 14 16 4
%e 15 25 2
%e 16 32 2
%e 17 289 2
%e 18 24 3
%e 19 361 2
%e 20 25 2
%e (End)
%t Table[Which[PrimeQ@ n, n^2, PrimePowerQ@ n, Block[{p = 2, e}, While[Set[e, IntegerExponent[n, p]] == 0, p = NextPrime@ p]; p^(e + 1)], True, Block[{k = n + 1}, While[PowerMod[n, k, k] != 0, k++]; k]], {n, 2, 61}] (* _Michael De Vlieger_, Jul 02 2017 *)
%o (PARI) \\ See Links section.
%Y Cf. A001221, A007947, A020639, A079277.
%K nonn
%O 2,1
%A _Rémy Sigrist_, Jul 01 2017
|
