A143907 If n = product{primes p(k)|n} p(k)^b(n,p(k)), where p(k) is the k-th prime that divides n (when these primes are listed from smallest to largest) and each b(n,p(k)) is a positive integer, then the sequence contains the non-prime-powers n such that p(k)^b(n,p(k)) < p(k+1) for all k, 1<=k<= -1 + number of distinct prime divisors of n.
6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 57, 58, 62, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124
Offset: 1
Keywords
Examples
2900 is factored as 2^2 * 5^2 * 29^1. Since 2^2 < 5 and 5^2 < 29, then 2900 is in the sequence. On the other hand, 60 is factored as 2^2 * 3^1 * 5^1. Even though 3^1 < 5, 2^2 is not < 3. So 60 is not in the sequence.
Programs
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Mathematica
okQ[n_] := With[{f = FactorInteger[n]}, If[Length[f] == 1, Return[False]]; For[i = 1, i < Length[f], i++, If[f[[i, 1]]^f[[i, 2]] >= f[[i+1, 1]], Return[False]]]; True]; Select[Range[200], okQ] (* Jean-François Alcover, May 16 2017, adapted from PARI *) -
PARI
isok(n) = {my(f = factor(n)); if (#f~ == 1, return (0)); for (i=1, #f~ - 1, if (f[i, 1]^f[i, 2] >= f[i+1, 1], return (0));); return (1);} \\ Michel Marcus, Jan 19 2014
Extensions
Extended by Ray Chandler, Nov 06 2008