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A100716
Numbers k such that p^p divides k for some prime p.
30
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
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. 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
KEYWORD
nonn
AUTHOR
Leroy Quet, Dec 10 2004
EXTENSIONS
More terms from Robert G. Wilson v, Dec 14 2004
STATUS
approved