A100716 Numbers k such that p^p divides k for some prime p.

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204, 208, 212

Offset: 1

Author

Leroy Quet, Dec 10 2004

Keywords

• nonn

Comments

Complement of A048103; A129251(a(n)) > 0; A051674 is a subsequence; A129254 = (terms a(k) such that a(k+1)=a(k)+1). - Reinhard Zumkeller, Apr 07 2007

A027748(a(n),k) <= A124010(a(n),k) for some k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012

Examples

			54 is included because 3^3 divides 54.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Complement: A048103.

Positions of nonzeros in A129251.

Cf. A100717, A129150, A129152.

Cf. A054744.

Cf. A051674 (a subsequence).

Subsequence of A276079 from which it differs for the first time at n=175, where a(175) = 628, while A276079(175) = 625, a value missing from here.

Programs

Haskell

a100716 n = a100716_list !! (n-1)
a100716_list = filter (\x -> or $
   zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012
(Scheme, with Antti Karttunen's IntSeq-library)
(define A100716 (NONZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016

Mathematica

fQ[n_] := Union[ Table[ #[[1]] <= #[[2]]] & /@ FactorInteger[n]][[ -1]] == True; Select[ Range[2, 215], fQ[ # ] &] (* Robert G. Wilson v, Dec 14 2004 *)
f[n_] := Module[{aux=FactorInteger[n]}, Last@Union@Table[aux[[i,1]] <=  aux[[i,2]], {i,Length[aux]}] == True]; Select[Range[2,215], f] (* José María Grau Ribas, Jan 25 2012 *)
Rest@ Select[Range@ 216, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] == 0 &] (* Michael De Vlieger, Aug 19 2016 *)

PARI

is(n)=forprime(p=2,default(primelimit),if(n%p^p==0,return(1));if(p^p>n,return(0))) \\ Charles R Greathouse IV, Jan 24 2012

Python

from itertools import count, islice
from sympy import factorint
def A100716_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:any(map(lambda d:d[1]>=d[0],factorint(n).items())),count(max(startvalue,1)))
A100716_list = list(islice(A100716_gen(),30)) # Chai Wah Wu, Jan 05 2023

Formula

a(n) ~ k*n with k = 1/(1 - Product(1 - p^-p)) = 3.5969959469... where the product is over all primes p. - Charles R Greathouse IV, Jan 24 2012

Extensions

More terms from Robert G. Wilson v, Dec 14 2004