A212164 Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k)).

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Author

Matthew Vandermast, May 22 2012

Keywords

• nonn
• easy

Examples

			40 = 2^3*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (namely, 3 and 1, although the 1 is often left implicit).   2 is less than the maximal exponent in 40's prime factorization, which is 3. Therefore, 40 belongs to the sequence.
		

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 A212167.

See also A212165, A212166, A212168.

Cf. A001221, A050326, A051903, A225230.

Subsequence of A188654.

Programs

Haskell

import Data.List (elemIndices)
a212164 n = a212164_list !! (n-1)
a212164_list = map (+ 1) $ findIndices (< 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

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 08 2024

Formula

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