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)).

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, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104

Offset: 1

Author

Matthew Vandermast, May 22 2012

Keywords

• nonn
• easy

Comments

Union of A212164 and A212166. Includes numerous subsequences that are subsequences of neither A212164 nor A212166.

Includes all factorials except A000142(3) = 6.

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

Examples

			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.
		

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 A212168.

See also A212167.

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.

Cf. A001221, A051903, A188654, A225230.

Programs

Haskell

import Data.List (findIndices)
a212165 n = a212165_list !! (n-1)
a212165_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. - Reinhard Zumkeller, May 03 2013