OFFSET
1,1
COMMENTS
First differs from A336568 in lacking 420.
Also distinct radicals of all composites k such that rad(k) = sopfr(k). For two distinct primes p and q, the equation p*q = x*p + y*q has no integer solutions with x, y >= 1: The left side is divisible by p and q, so x*p + y*q can equal p*q only if y is a multiple of p and x is a multiple of q, which makes the right side at least 2*p*q. - Felix Huber, Dec 22 2025
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = k*n + O(n log log n/log n), where k = Pi^2/6. - Charles R Greathouse IV, Dec 22 2025
EXAMPLE
The terms and their prime indices begin:
30: {1,2,3} 182: {1,4,6} 285: {2,3,8}
42: {1,2,4} 186: {1,2,11} 286: {1,5,6}
66: {1,2,5} 190: {1,3,8} 290: {1,3,10}
70: {1,3,4} 195: {2,3,6} 310: {1,3,11}
78: {1,2,6} 210: {1,2,3,4} 318: {1,2,16}
102: {1,2,7} 222: {1,2,12} 322: {1,4,9}
105: {2,3,4} 230: {1,3,9} 330: {1,2,3,5}
110: {1,3,5} 231: {2,4,5} 345: {2,3,9}
114: {1,2,8} 238: {1,4,7} 354: {1,2,17}
130: {1,3,6} 246: {1,2,13} 357: {2,4,7}
138: {1,2,9} 255: {2,3,7} 366: {1,2,18}
154: {1,4,5} 258: {1,2,14} 370: {1,3,12}
165: {2,3,5} 266: {1,4,8} 374: {1,5,7}
170: {1,3,7} 273: {2,4,6} 385: {3,4,5}
174: {1,2,10} 282: {1,2,15} 390: {1,2,3,6}
MAPLE
A350352 := proc(n) local k; option remember; if n = 1 then 30; else for k from A350352(n - 1) + 1 do if NumberTheory:-Radical(k) = k and 2 < NumberTheory:-Omega(k, 'distinct') then return k; end if; end do; end if; end proc: seq(A350352(n), n = 1 .. 54); # Felix Huber, Dec 22 2025
MATHEMATICA
Select[Range[100], SquareFreeQ[#]&&PrimeOmega[#]>=3&]
PROG
(Python)
from sympy import factorint
def ok(n): f = factorint(n, multiple=True); return len(f) == len(set(f)) > 2
print([k for k in range(431) if ok(k)]) # Michael S. Branicky, Jan 14 2022
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A350352(n):
def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) for d in g(x, a2, b2, c*b2, m-1)))
def f(x): return int(n+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, i)) for i in range(3, x.bit_length())))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
return bisection(f, n, n) # Chai Wah Wu, Sep 11 2024
(PARI) is(n, f=factor(n))=my(e=f[, 2]); #e>2 && vecmax(e)==1 \\ Charles R Greathouse IV, Jul 08 2022
(PARI) list(lim)=my(v=List()); forsquarefree(n=30, lim\1, if(#n[2][, 2]>2, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 08 2022
CROSSREFS
This is the squarefree case of A033942.
Including squarefree semiprimes gives A120944.
The squarefree complement consists of 1 and A167171.
These are the Heinz numbers of the partitions counted by A347548.
A000040 lists prime numbers (exactly 1 prime factor).
A005117 lists squarefree numbers.
A006881 lists squarefree numbers with exactly 2 prime factors.
A007304 lists squarefree numbers with exactly 3 prime factors.
A046386 lists squarefree numbers with exactly 4 prime factors.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 11 2022
STATUS
approved