A212166 Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)).

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Author

Matthew Vandermast, May 22 2012

Keywords

• nonn
• easy

Examples

			36 = 2^2*3^2 has 2 positive exponents in its prime factorization. The maximal exponent in its prime factorization is also 2. Therefore, 36 belongs to this 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

Includes subsequences A000040, A006939, A138534, A181555, A181825.

See also A212164, A212165, A212167, A212168.

Cf. A001221, A050326, A051903, A188654 (complement), A225230.

Programs

Haskell

import Data.List (elemIndices)
a212166 n = a212166_list !! (n-1)
a212166_list = map (+ 1) $ elemIndices 0 a225230_list
-- Reinhard Zumkeller, May 03 2013

Mathematica

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] == Length[f]]; Select[Range[424], 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)) = 1. - Reinhard Zumkeller, May 03 2013