OFFSET
1,1
COMMENTS
The asymptotic density of this sequence is 1 - Product_{i>=1} 1-prime(i)^(-1-i) = 0.2789766... - Amiram Eldar, Oct 21 2020
These are numbers with a prime index that is less than its own multiplicity. First differs from A387577 in having 16807. - Gus Wiseman, Sep 16 2025
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..5000
EXAMPLE
625 = 5*5*5*5 = prime(3)^4 so it is divisible by prime(3)^(3+1), and thus 625 is included in the sequence.
From Gus Wiseman, Sep 14 2025 (Start):
The prime indices of 270 are y = {1,2,2,2,3}, and we have more than 2 appearances of 2, so 270 is in the sequence.
The prime indices of 450 are y = {1,2,2,3,3}, in which no part k appears more than k times, so 450 is not in the sequence.
(End)
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Or@@Table[Count[prix[#], k]>k, {k, Union[prix[#]]}]&] (* Gus Wiseman, Sep 14 2025 *)
PROG
(Scheme) ;; With Antti Karttunen's IntSeq-library.
(Python)
from sympy import primepi, isprime, primefactors, factorint
def a028234(n):
f=factorint(n)
minf = min(f)
return 1 if n==1 else n//(minf**f[minf])
def a067029(n):
f=factorint(n)
return 0 if n==1 else f[min(f)]
def a049084(n): return primepi(n) if isprime(n) else 0
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def a(n): return 0 if n==1 else a(a028234(n)) + (1 if a067029(n) > a055396(n) else 0)
print([n for n in range(1, 301) if a(n)!=0]) # Indranil Ghosh, Jun 21 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 18 2016
STATUS
approved
