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)).

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, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Author

Matthew Vandermast, May 22 2012

Keywords

• nonn

Comments

Union of A212166 and A212168. Includes numerous subsequences that are subsequences of neither A212166 nor A212168.

Examples

			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.
		

References

• 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.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Primefan, The First 2500 Integers Factored (first of 5 pages).

Crossrefs

Complement of A212164. See also A212165.

Subsequences (none of which are subsequences of A212166 or A212168) include A002110, A051451, A129912, A179983, A181826, A181827, A182862, A182863. Includes all members of A003418.

Cf. A001221, A050326, A051903, A188654, A225230.

Programs

Haskell

import Data.List (findIndices)
a212167 n = a212167_list !! (n-1)
a212167_list = map (+ 1) $ findIndices (>= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

Maple

isA212167 := proc(n)
    simplify(A051903(n) <= A001221(n)) ;
end proc:
for n from 1 to 1000 do
    if isA212167(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 06 2021

Mathematica

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] <= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

PARI

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) <= #e; } \\ Amiram Eldar, Sep 09 2024

Formula

A225230(a(n)) >= 0; A050326(a(n)) > 0. - Reinhard Zumkeller, May 03 2013