cp's OEIS Frontend

A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).

%I A212167 #22 Sep 09 2024 02:41:14
%S A212167 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,26,28,29,30,31,33,
%T A212167 34,35,36,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,
%U A212167 62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83
%N A212167 Numbers k such that the maximum exponent in its prime factorization is not greater than the number of positive exponents (A051903(k) <= A001221(k)).
%C A212167 Union of A212166 and A212168. Includes numerous subsequences that are subsequences of neither A212166 nor A212168.
%D A212167 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 A212167 Reinhard Zumkeller, <a href="/A212167/b212167.txt">Table of n, a(n) for n = 1..10000</a>
%H A212167 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 A212167 Primefan, <a href="http://primefan.tripod.com/500factored.html">The First 2500 Integers Factored</a> (first of 5 pages).
%F A212167 A225230(a(n)) >= 0; A050326(a(n)) > 0. - _Reinhard Zumkeller_, May 03 2013
%e A212167 40 = 2^3*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although the 1 is often left implicit).  2 is less than the maximal exponent in 40's prime factorization, which is 3. Therefore, 40 does not belong to the sequence. But 10 = 2^1*5^1 and 20 = 2^2*5^1 belong, since the maximal exponents in their prime factorizations are 1 and 2 respectively.
%p A212167 isA212167 := proc(n)
%p A212167     simplify(A051903(n) <= A001221(n)) ;
%p A212167 end proc:
%p A212167 for n from 1 to 1000 do
%p A212167     if isA212167(n) then
%p A212167         printf("%d,",n) ;
%p A212167     end if;
%p A212167 end do: # _R. J. Mathar_, Jan 06 2021
%t A212167 okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] <= Length[f]]; Select[Range[1000], okQ] (* _T. D. Noe_, May 24 2012 *)
%o A212167 (Haskell)
%o A212167 import Data.List (findIndices)
%o A212167 a212167 n = a212167_list !! (n-1)
%o A212167 a212167_list = map (+ 1) $ findIndices (>= 0) a225230_list
%o A212167 -- _Reinhard Zumkeller_, May 03 2013
%o A212167 (PARI) is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) <= #e; } \\ _Amiram Eldar_, Sep 09 2024
%Y A212167 Complement of A212164. See also A212165.
%Y A212167 Subsequences (none of which are subsequences of A212166 or A212168) include A002110, A051451, A129912, A179983, A181826, A181827, A182862, A182863. Includes all members of A003418.
%Y A212167 Cf. A001221, A050326, A051903, A188654, A225230.
%K A212167 nonn
%O A212167 1,2
%A A212167 _Matthew Vandermast_, May 22 2012