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

1,1

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

Links

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

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

Example

54 is included because 3^3 divides 54.

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 *)

Prog

(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
(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
(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

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.

Sequence in context: A285466 A274141 A086133 * A328251 A276079 A311124

Adjacent sequences: A100713 A100714 A100715 * A100717 A100718 A100719

Keyword

nonn

Author

Leroy Quet, Dec 10 2004

Extensions

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

Status

approved