A212168 - OEIS

A212168

Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142, 143

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

OFFSET

1,1

COMMENTS

A225230(a(n)) > 1; A050326(a(n)) > 1. - Reinhard Zumkeller, May 03 2013

Subsequence of A130092. - Ivan N. Ianakiev, Sep 17 2019

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)

EXAMPLE

10 = 2^1*5^1 has 2 distinct prime factors, hence 2 positive exponents in its prime factorization (although the 1s are often left implicit). 2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 belongs to the sequence.

MATHEMATICA

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

Select[Range[200], Max[FactorInteger[#][[All, 2]]]<PrimeNu[#]&] (* Harvey P. Dale, Nov 21 2018 *)

PROG

(Haskell)

import Data.List (findIndices)

a212168 n = a212168_list !! (n-1)

a212168_list = map (+ 1) $ findIndices (> 0) a225230_list

-- Reinhard Zumkeller, May 03 2013

(PARI) is(n, f=factor(n))=my(e=f[, 2]); #e && vecmax(e)<#e \\ Charles R Greathouse IV, Jan 09 2022

CROSSREFS

Complement of A212165. See also A212164, A212166-A212167.