A129254 -id:A129254 - cp's OEIS frontend

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

Leroy Quet, Dec 10 2004

nonn

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

			54 is included because 3^3 divides 54.

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

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.

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

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 *)

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

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

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

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

A376469 Starts of runs of 3 consecutive integers in which each member of the run has at least one divisor of the form p^e with p <= e, where p is a prime.

Original entry on oeis.org

71874, 109375, 156248, 181250, 228123, 265624, 409374, 446875, 493748, 518750, 565623, 603124, 746874, 784375, 831248, 856250, 903123, 940624, 1084374, 1121875, 1168748, 1193750, 1240623, 1278124, 1421874, 1459375, 1506248, 1531250, 1578123, 1615624, 1759374, 1796875

Offset: 1

Amiram Eldar, Sep 23 2024

nonn

The start of the least run of 4 (and also 5) consecutive integers with this property is 3988418748.

The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 18, 178, 1783, 17845, 178458, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00001784... .

			71874 = 2 * 3^3 * 11^3 is a term since it is divisible by 3^3, 71875 = 5^5 * 23 is divisible by 5^5, and 71876 = 2^2 * 7 * 17 * 151 is divisible by 2^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A100716, A070258 and A129254.

q[n_] := q[n] = AnyTrue[FactorInteger[n], First[#] <= Last[#] &]; Select[Range[2*10^6], q[#] && q[#+1] && q[#+2] &]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,1] <= f[i,2], return(1))); 0;}
lista(kmax) = {my(is1 = 0, is2 = 0, is3); for(k = 3, kmax, is3 = is(k); if(is1 && is2 && is3, print1(k-2, ", ")); is1 = is2; is2 = is3);}

cp's OEIS Frontend

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

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