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
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
Programs
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Haskell
import Data.List (findIndices) a212165 n = a212165_list !! (n-1) a212165_list = map (+ 1) $ findIndices (<= 0) a225230_list -- Reinhard Zumkeller, May 03 2013
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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
Comments