A212165 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

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).

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(Haskell)

import Data.List (findIndices)

a212165 n = a212165_list !! (n-1)

a212165_list = map (+ 1) $ findIndices (<= 0) a225230_list

-- Reinhard Zumkeller, May 03 2013

(PARI) is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) >= #e; } \\ Amiram Eldar, Sep 08 2024

CROSSREFS

Complement of A212168.