A048103 Numbers not divisible by p^p for any prime p.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98

Offset

1,2

Comments

If a(n) = Product p_i^e_i then p_i > e_i for all i.

Complement of A100716; A129251(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2007

Density is 0.72199023441955... = Product_{p>=2} (1 - p^-p) where p runs over the primes. - Charles R Greathouse IV, Jan 25 2012

A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). - Reinhard Zumkeller, Apr 28 2012

Range of A276086. Also numbers not divisible by m^m for any natural number m > 1. - Antti Karttunen, Nov 18 2024

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) ~ kn with k = 1/Product_{p>=2}(1 - p^-p) = Product_{p>=2}(1 + 1/(p^p - 1)) = 1.3850602852..., where the product is over all primes p. - Charles R Greathouse IV, Jan 25 2012

For n >= 1, A377982(a(n)) = n. - Antti Karttunen, Nov 18 2024

Example

6 = 2^1 * 3^1 is OK but 12 = 2^2 * 3^1 is not.
625 = 5^4 is present because it is not divisible by 5^5.

Mathematica

{1}~Join~Select[Range@ 120, Times @@ Boole@ Map[First@ # > Last@ # &, FactorInteger@ #] > 0 &] (* Michael De Vlieger, Aug 19 2016 *)

Prog

(Haskell)
a048103 n = a048103_list !! (n-1)
a048103_list = filter (\x -> and $
   zipWith (>) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012
(Scheme, with Antti Karttunen's IntSeq-library)
(define A048103 (ZERO-POS 1 1 A129251))
;; Antti Karttunen, Aug 18 2016
(PARI) isok(n) = my(f=factor(n)); for (i=1, #f~, if (f[i, 1] <= f[i, 2], return(0))); return(1); \\ Michel Marcus, Nov 13 2020
(PARI) A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ (A359550 is the characteristic function for A048103) - Antti Karttunen, Nov 18 2024
(Python)
from itertools import count, islice
from sympy import factorint
def A048103_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda d:d[1]<d[0], factorint(n).items())), count(max(startvalue, 1)))
A048103_list = list(islice(A048103_gen(), 30)) # Chai Wah Wu, Jan 05 2023

Crossrefs

Complement: A100716.

Positions of 0's in A129251, A342023, A376418, positions of 1's in A327936, A342007, A359550 (characteristic function).

Cf. A048102, A048104, A051674 (p^p), A054743, A054744, A377982 (a left inverse, partial sums of char. fun, see also A328402).

Cf. A276086 (permutation of this sequence, see also A376411, A376413).

Subsequences: A002110, A005117, A006862, A024451 (after its initial 0), A057588, A099308 (after its initial 0), A276092, A327934, A328832, A359547, A370114, A371083, A373848, A377871, A377992.

Also A276078 is a subsequence, from which this differs for the first time at n=451 where a(451)=625, while that value is missing from A276078.

Sequence in context: A195291 A042968 A337037 * A276078 A193303 A285465

Adjacent sequences: A048100 A048101 A048102 * A048104 A048105 A048106

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Apr 22 2000

Status

approved