cp's OEIS Frontend

A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

%I A212165 #25 Sep 08 2024 15:16:11
%S A212165 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,24,25,27,28,29,31,32,36,
%T A212165 37,40,41,43,44,45,47,48,49,50,52,53,54,56,59,61,63,64,67,68,71,72,73,
%U A212165 75,76,79,80,81,83,88,89,92,96,97,98,99,100,101,103,104
%N A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).
%C A212165 Union of A212164 and A212166.  Includes numerous subsequences that are subsequences of neither A212164 nor A212166.
%C A212165 Includes all factorials except A000142(3) = 6.
%C A212165 Observation: all terms in DATA section are also the first 65 numbers n whose difference between the arithmetic derivative of n and the sum of the divisors of n is nonnegative. - _Omar E. Pol_, Dec 19 2012
%D A212165 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
%H A212165 Reinhard Zumkeller, <a href="/A212165/b212165.txt">Table of n, a(n) for n = 1..10000</a>
%H A212165 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A212165 Primefan, <a href="http://primefan.tripod.com/500factored.html">The First 2500 Integers Factored</a> (first of 5 pages).
%F A212165 A225230(a(n)) <= 0. - _Reinhard Zumkeller_, May 03 2013
%e A212165 10 = 2^1*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although 1s are often left implicit).  2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 does not belong to the sequence. But 20 = 2^2*5^1 and 40 = 2^3*5^1 belong, since the largest exponents in their prime factorizations are 2 and 3 respectively.
%t A212165 okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; Select[Range[1000], okQ] (* _T. D. Noe_, May 24 2012 *)
%o A212165 (Haskell)
%o A212165 import Data.List (findIndices)
%o A212165 a212165 n = a212165_list !! (n-1)
%o A212165 a212165_list = map (+ 1) $ findIndices (<= 0) a225230_list
%o A212165 -- _Reinhard Zumkeller_, May 03 2013
%o A212165 (PARI) is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) >= #e;} \\ _Amiram Eldar_, Sep 08 2024
%Y A212165 Complement of A212168.
%Y A212165 See also A212167.
%Y A212165 Subsequences (none of which are subsequences of A212164 or A212166) include A000079, A001021, A066120, A087980, A130091, A141586, A166475, A181818, A181823, A181824, A182763, A212169. Also includes all terms in A181813 and A181814.
%Y A212165 Cf. A001221, A051903, A188654, A225230.
%K A212165 nonn,easy
%O A212165 1,2
%A A212165 _Matthew Vandermast_, May 22 2012